approximate syllogistic reasoning: a … · de la silogística aristotélica cabe destacar que la...

UNIVERSIDADE DE SANTIAGO DE COMPOSTELA

Departamento de Electrónica e Computación

Tese de doutoramento

APPROXIMATE SYLLOGISTIC REASONING: ACONTRIBUTION TO INFERENCE PATTERNS AND USE CASES

Presentada por:Martín Pereira Fariña

Dirigida por:Alberto J. Bugarín DizAlejandro Sobrino Cerdeiriña

Santiago de Compostela, Novembro 2013

Alberto J. Bugarín Diz, Catedrático de Universidade da Área de Ciencia da Computación e

Intelixencia Artificial da Universidade de Santiago de Compostela

Alejandro Sobrino Cerdeiriña, Profesor Titular da Área de Lóxica e Filosofía da Ciencia da

Universidade de Santiago de Compostela

FAN CONSTAR:

Que a memoria titulada APPROXIMATE SYLLOGISTIC REASONING: A CONTRIBUTIONTO INFERENCE PATTERNS AND USE CASES ven de ser realizada por D. Martín Pereira Fariñabaixo a nosa dirección no Departamento de Electrónica e Computación da Universidade de Santiago de

Compostela, e constitúe a Tese que presenta para optar ao grao de Doutor.


Alberto J. Bugarín DizCodirector da tese

Alejandro Sobrino CerdeiriñaCodirector da tese

Martín Pereira FariñaAutor da tese

Á miña familia e a Noa,polo seu constante apoio

¿Cantas veces lle teño dito que se eliminamos o

imposible, o que queda, por improbable que pareza,

ten que ser a verdade?

Sherlock Holmes, O Signo dos Catro

Agradecementos

Unha vez rematado este longo e duro traballo pero á súa vez estimulante e enriquecedor que éfacer unha tese de doutoramento, é de xustiza dedicar as primeiras liñas da mesma a expresaro meu máis sincero agradecemento a todas aquelas persoas e institucións que fixeron posibleo seu desenvolvemento.

Quero encomezar polos codirectores da tese, os profesores Alberto José Bugarín Diz eAlejandro Sobrino Cerdeiriña, quen seguiron este traballo dende o primeiro ata o último día;sen a súa confianza e axuda esta investigación non tería sido posible. Tamén quero agradecerespecialmente a axuda dos doutores Félix Díaz Hermida e Juan Carlos Vidal Aguiar, poissen a súa colaboración este volume tampouco tería visto a luz. Por último, non podo deixarde mostrar o meu agradecemento a Alejandro Ramos e a Pedro Montoto, pois colaboramoscóbado con cóbado no desenvolvemento da libraría software.

Tamén quero expresar o meu agradecemento ao Departamento de Electrónica e Compu-tación, principalmente ao Grupo de Sistemas Intelixentes, ao de Lóxica e Filosofía Moral eao Centro Singular de Investigación en Tecnoloxías da Información (CITIUS), pois foi no seuseo e baixo os seus teitos nos que desenvolvín a maior parte do meu traballo. Tampouco podoesquecerme do European Centre for Soft Computing nin da xente que alí traballa, pois duranteo outono do 2011 pasei entre eles unha tremendamente agradable estadía durante a cal puidenmanter longas conversas co doutor Gracián Triviño Barros e das que xurdiron frutíferas ideas;nin da Universidade Técnica de Viena, pois durante o outono do 2012 acolléronme moi aga-rimosamente para unha nova estadía durante a cal se me brindou a oportunidade de coñecero traballo do Prof. Dipl.-Ing. Dr. techn. Christian Georg Fermüller e do seu doutorando ChrisRoschger.

Tampouco podo esquecerme dos vellos e novos amigos labrados durante estes anos, poiseles fixeron moito máis levadía esta tarefa. E, por suposto, agradecer o apoio da miña familia

x

e de Noa, que dende sempre me apoiaron para que acadara os meus soños e metas con plenaconfianza no meu criterio.

Para rematar, non podo deixar de mostrar o meu agradecemento ao apoio económico reci-bido durante estes anos. En primeiro lugar, destacar o soporte ofrecido pola Xunta de Galiciaa través das súas axudas para a “Realización de estudos de terceiro ciclo nas universidades dosistema universitario de Galicia” e polo Ministerio de Educación, Cultura y Deporte a travésdas bolsas do seu programa “Formación del Profesorado Universitario (FPU)”. Ao Ministeriode Economía y Competitividad, polo soporte recibido a través do proxecto “Descubrimientoautomático de reglas borrosas no convencionales: Bases de Conocimiento Temporal Borrosoy TSK de Estructura Variable” (TIN2008-00040) e ao mesmo ministerio xunto cos FondosEuropeos para o Desenvolvemento Rexional (ERDF/FEDER) tamén polo proxecto “Descrip-ción Lingüística de Fenómenos Complejos: Cuantificadores Borrosos Generalizados en Pro-posiciones Temporales” (TIN2011-29827-C02-02); de novo aos fondos ERDF/FEDER, nestecaso en colaboración coa Xunta de Galicia, polos proxectos CN2012/151 e CN2011/058; aoGrupo de Sistemas Intelixentes e ao CITIUS polo seu apoio económico á hora de realizar asestadías e asistir a congresos.


Contents

Resumen de la tesis 1

Preface 9

1 State of the Art in Syllogistic Reasoning 191.1 Set-Based Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.1.1 Aristotelian Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . 211.1.2 Intermediate Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . 331.1.3 Exception Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . . 421.1.4 Interval Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . . . 511.1.5 Fuzzy Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 651.1.6 Generalized Intermediate Syllogistics . . . . . . . . . . . . . . . . . 74

1.2 Conditional Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801.2.1 Probabilistic Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . 821.2.2 Support Logic Syllogistics . . . . . . . . . . . . . . . . . . . . . . . 901.2.3 Qualified Syllogistics . . . . . . . . . . . . . . . . . . . . . . . . . . 93

1.3 Discussion: Set-Based I. vs. Conditional I. . . . . . . . . . . . . . . . . . . . 96

2 Syllogistic Reasoning Using the TGQ 1012.1 Theory of Categorical Statements Using the TGQ . . . . . . . . . . . . . . . 1022.2 Theory of Syllogism Using the TGQ . . . . . . . . . . . . . . . . . . . . . . 104

2.2.1 Inference Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.2.2 Division of the Universe in Disjoint Sets . . . . . . . . . . . . . . . . 1082.2.3 Transformation into Inequations . . . . . . . . . . . . . . . . . . . . 1092.2.4 Definition and Resolution of the Equivalent Optimization Problem . . 111

xii Contents

2.3 Some Examples of Syllogisms Using the TGQ . . . . . . . . . . . . . . . . . 1132.3.1 Syllogisms with Crisp Interval Quantifiers . . . . . . . . . . . . . . . 1132.3.2 Syllogisms with Fuzzy Quantifiers Approximated as Pairs of Intervals 1172.3.3 Syllogisms with Fuzzy Quantifiers . . . . . . . . . . . . . . . . . . . 117

2.4 Combination of Absolute and Proportional Quantifiers . . . . . . . . . . . . 1252.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3 Approximate Chaining Syllogism 1293.1 Similarity as Fuzzy Inclusion, Fuzzy Overlapping and Synonymy . . . . . . . 130

3.1.1 Similarity as Fuzzy Inclusion . . . . . . . . . . . . . . . . . . . . . 1313.1.2 Similarity as Overlapping . . . . . . . . . . . . . . . . . . . . . . . 1323.1.3 Similarity as Synonymy . . . . . . . . . . . . . . . . . . . . . . . . 134

3.2 A Proposal for a Measurement of Synonymy between Two Terms . . . . . . 1353.2.1 Calculation of the Degree of Synonymy . . . . . . . . . . . . . . . . 138

3.3 Basic Inference Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.3.1 A Brief Analysis of the Measures of Synonymy based on Examples . 1453.3.2 Syllogisms with Weighted Quantified Statements . . . . . . . . . . . 149

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4 Use Cases: Probabilistic Reasoning Using Syllogisms 1534.1 Generalized Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.2 Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.2.1 Topological Types of Bayesian Networks . . . . . . . . . . . . . . . 1614.2.2 Types of Inferences with Bayesian Networks . . . . . . . . . . . . . 1644.2.3 BNs in Medical Diagnosis: Lung Cancer . . . . . . . . . . . . . . . 165

4.3 Probabilistic Reasoning in Legal Argumentation . . . . . . . . . . . . . . . . 1714.4 Qualitative Version using Linguistic labels . . . . . . . . . . . . . . . . . . . 1744.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Conclusions 177

Bibliography 185

List of Figures 193

Contents xiii

List of Tables 195

Resumen de la tesis

La investigación que presentamos en esta memoria de tesis doctoral se encuadra en el mar-co de la Computación con Palabras y del Razonamiento Aproximado. En ella abordamos eltratamiento de un tipo de inferencia deductiva, el silogismo, cuyo patrón de razonamientose basa en el encadenamiento de términos mediante el uso de oraciones cuantificadas de laforma Q S son P, donde Q designa un cuantificador binario, S el término sujeto y P el términopredicado.

En su aproximación clásica y nítida, desarrollada por Aristóteles [7], se contemplan argu-mentos compuestos por tres términos, dos premisas y una conclusión. De los tres términos,dos son los llamados extremos, que aparecen en la conclusión, y el otro es el término medio,que sirve de conexión o encadenamiento entre ellos y únicamente aparece en las premisas.Los cuantificadores considerados son los cuatro clásicos: “todos”, “ningún”, “alguno”, “al-guno. . . no” (organizados en el Cuadrado Lógico de la Oposición); por ello todo silogismo sereduce a una combinación de las cuatro oraciones generadas por estos cuantificadores y lasdistintas posiciones que puede ocupar el término medio en las premisas (las cuatro Figurasaristotélicas).

De la silogística aristotélica cabe destacar que la forma lógica de las proposiciones es suforma gramatical, por lo que la construcción de un silogismo resulta intuitiva y sencilla paracualquier usuario. Sin embargo, dado que se limita a sólo cuatro tipos de oraciones y argu-mentos de dos premisas, su capacidad expresiva también resulta muy restringida. Además,tampoco se consideran cuantificadores imprecisos o vagos como “pocos”, “muchos”, “casitodos”,. . . lo que limita todavía más su expresividad. Por otra parte, el número de términosaceptados en los argumentos (tres) también es insuficiente ya que, aunque los humanos noelaboramos argumentos muy largos, si pueden aparecer fácilmente algunos más.

2 Resumen de la tesis

La hipótesis de trabajo de nuestra investigación ha sido que los modelos existentes de ra-zonamiento silogístico sólo consideran una pequeña parte de los silogismos que los humanospodemos manejar, tanto desde el punto de vista de los cuantificadores como de los patrones deinferencia. Como explicaremos, estos siguen siendo bastante limitados y, por esta razón, he-mos establecido como principal propósito de nuestra investigación el desarrollo de un nuevomodelo de silogismo que contribuya a mejorar la capacidad de la Computación con Palabraspara tratar inferencias usando lenguaje natural en este contexto. Hemos definido siete objeti-vos (detallados en el prefacio de esta memoria de tesis), que en este resumen agrupamos entres: i) analizar los modelos silogísticos propuestos e identificar los puntos débiles a mejorarasí como los diversos aspectos a desarrollar; ii) definir un nuevo modelo para la representaciónde las oraciones cuantificadas y un nuevo patrón de inferencia que no sólo incluya cuantifi-cadores vagos sino también imprecisión en el encadenamiento; iii) analizar casos de uso delsilogismo no previamente considerados.

Análisis de diversos modelos de silogismo

En la bibliografía podemos encontrar diversas propuestas que tratan de superar las menciona-das limitaciones. Analizamos aquellas que se centran en introducir los cuantificadores vagoscomo “muchos”, “pocos”,. . . en los mismos (o prácticamente) patrones de inferencia. En fun-ción de cómo estos modelos interpretan las proposiciones, los clasificamos en dos categorías:a) interpretación basada en conjuntos; b) interpretación condicional.

La interpretación basada en conjuntos asume que los términos de las proposiciones deno-tan conjuntos de elementos y los cuantificadores una relación de cardinalidad entre ellos. Porejemplo, “todos los hombres son mortales” significa que el conjunto hombres está incluidototalmente en el conjunto de mortales. Describimos cinco modelos diferentes: la Interme-

diate Syllogistics, la Exception Syllogistics, la Interval Syllogistics, la Fuzzy Syllogistics y laGeneralized Intermediate Syllogistics.

La Intermediate Syllogistics [63] define un nuevo Cuadrado Lógico de la Oposición enel que se introducen nuevos cuantificadores, los llamados intermedios (por ejemplo, “mu-chos”,. . . ), que se sitúan en el espacio entre el universal y el existencial. La Exception Syllo-

gistics [39] destaca porque introduce los cuantificadores no proporcionales, en particular losllamados de excepción (por ejemplo, “todos menos tres”,. . . ), y una reescritura de los pro-porcionales en estos términos. Este modelo también admite la conclusión múltiple, lo que le


da un cariz heurístico a este tipo de inferencias (hay varias conclusiones válidas posibles).Sin embargo, ambos modelos se limitan a las figuras aristotélicas y a usar definiciones nítidaspara los cuantificadores.

Los modelos Interval Syllogistics [18], Fuzzy Syllogistics [93] y Generalized Intermediate

Syllogistics [37] pertenecen a la lógica borrosa. Todos comparten como objetivo el modeladode los cuantificadores imprecisos mediante conjuntos borrosos aunque con diferentes apro-ximaciones. La Interval Syllogistics modela los cuantificadores mediante sub-intervalos delintervalo [0,1] e interpreta el proceso de razonamiento como un proceso de optimización,donde las premisas del silogismo constituyen un conjunto de restricciones y la conclusión elmáximo intervalo compatible con ellas. La Fuzzy Syllogistics identifica los cuantificadorescon números borrosos (proporcionales cuando los cuantificadores son proporcionales y abso-lutos cuando son absolutos) obtenidos de aplicar la medida de cardinalidad borrosa Σ−count

a los términos e interpreta el proceso de razonamiento aplicando el Principio de Extensión deCuantificadores y la aritmética borrosa. Por último, en la Generalized Intermediate Syllogis-

tics se propone una generalización de la Intermediate Syllogistics en el marco de una teoríade tipos borrosos pero usando definiciones borrosas para los cuantificadores.

La interpretación condicional, por su parte, asume que una oración cuantificada está ex-presando, en su estructura profunda, una oracional condicional donde el cuantificador denotael grado de fortaleza de la conexión entre el antecedente y el consecuente. Por ejemplo, “to-dos los hombres son mortales” significa que para cualquier elemento x del universo, “si x

tiene la propiedad de ser hombre, entonces x es mortal con un grado de fortaleza 1”. Loscuantificadores borrosos, en este caso, denotan un vínculo más débil.

Nosotros analizamos tres modelos: Probabilistic Syllogistics [43], Support Logic Syllogis-

tics [79] y Qualified Syllogistics [71]. La Probabilistic Syllogistics, desarrollada en el ámbitode la psicología del razonamiento, asume que los cuantificadores expresan realmente proba-bilidad; por ejemplo, “todos” significa “100%” o “casi todos” es “más del 90%”. Desde elpunto de vista de los patrones de inferencia, aunque se limita a analizar las cuatro Figurasaristotélicas, asimila el razonamiento a un proceso heurístico aunque a veces nos conduzcaa conclusiones erróneas. La Support Logic Syllogistics sustituye la probabilidad por la teoríade la evidencia de Dempster-Shafer; así pues, un cuantificador viene a denotar el grado deconfianza del usuario en la relación condicional. Sobre los patrones de inferencia, conside-ra sólo algunos de la Fuzzy Syllogistics pero obteniendo resultados diferentes. Por último, laQualified Syllogistics establece que los cuantificadores pueden denotar cantidad, frecuencia o


probabilidad puesto que existe cierta equivalencia entre estos conceptos; por ejemplo, “casitodos”≡ “casi siempre”≡ “casi seguro”. Respecto de los patrones de inferencia, simplementeasume el Modus Ponens sin ningún tipo de referencia al patrón de silogismo típico.

A partir del análisis de estos modelos hemos establecido los puntos a desarrollar en nues-tro modelo de silogismo. Varios de los modelos analizados tratan los cuantificadores borrososmediantes definiciones nítidas; aquellos que efectivamente usan definiciones borrosas no pue-den tratar correctamente los modos clásicos. Además, casi todos los modelos consideraronúnicamente cuantificadores proporcionales, descartando otros definidos por la Teoría de losCuantificadores Generalizados (TGQ) como los absolutos, comparativos,. . . que también sonútiles para hacer inferencias. Por otra parte, los patrones de inferencia propuestos se limitaronprácticamente (excepto en los modelos borrosos) a las cuatro figuras aristotélicas. Por últi-mo, los patrones heurísticos, que permiten más inferencias que las deducciones, o bien sóloconsideraban los patrones aristotélicos (como la Probabilistic Syllogistics) o no lo manejabancorrectamente (como la Interval Syllogistics).

Desarrollo de nuestro nuevo modelo de silogismo

A partir de las conclusiones anteriores definimos tres puntos a satisfacer por nuestro nuevomodelo de silogismo [56]: i) incluir todos los tipos de cuantificadores binarios definidos por laTGQ; ii) desarrollar un patrón de inferencia que permita más inferencias que las deductivas;iii) modelar argumentos con cualquier número de términos y premisas sin necesidad de seguirun patrón previamente predefinido.

Adoptamos la interpretación basada en conjuntos, pues es compatible con la TGQ, nospermite el tratamiento de cuantificadores no proporcionales y puede ser gráficamente repre-sentada de forma intuitiva y simple mediante diagramas de Venn. Respecto del patrón deinferencia, tomamos como punto de partida la Interval Syllogistics [18] pues, a pesar de noconsiderar el caso clásico, si es compatible con él, considera cuantificadores vagos y preci-sos y nos permite adoptar una perspectiva heurística al interpretar el razonamiento como unproceso de optimización matemática.

Nuestro primer paso en el desarrollo del modelo ha sido la definición de las oracionescuantificadas según la TGQ. Así pues, cualquier oración cuantificada binaria se expresa me-diante una ecuación entre los términos de la proposición y la cardinalidad denotada por elcuantificador. Por lo tanto, para cualquier tipo de cuantificador binario de la TGQ obtene-


mos una expresión que se refiere unívocamente a una región del universo del discurso. Porejemplo, en el caso de los cuantificadores precisos, “quince estudiantes son altos” se definecomo |estudiantes| ∩ |altos| = 15, donde |estudiantes| denota la cardinalidad del conjuntoestudiantes y |altos| la cardinalidad del de personas altas; en los cuantificadores borrosos,“pocos estudiantes son altos” se define como |estudiantes|∩|altos|

|estudiantes| = pocos.El siguiente paso fue el patrón de inferencia. Siguiendo la Interval Syllogistics, las premi-

sas del argumento se interpretan como un conjunto de restricciones y la conclusión como elvalor que tiene que ser máximamente compatible con esas restricciones. Así pues, transfor-mamos el problema de razonamiento en un problema de optimización, dividiéndolo en tresetapas: i) división del universo en conjuntos disjuntos; ii) definición de las proposiciones co-mo sistemas de inecuaciones; iii) aplicación del método de optimización necesario en cadacaso para resolver el problema del razonamiento.

En el paso i), apoyándonos en los diagramas de Venn, identificamos todos los conjuntosdisjuntos de los términos del silogismo. Cada premisa del argumento designa a un subconjuntoo a una operación booleana entre ellos y el cuantificador la cardinalidad correspondiente. Porlo tanto, cada una de las premisas genera un sistema de inecuaciones.

En el paso ii) se organizan las inecuaciones obtenidas en el paso i). Si los cuantificadoresson precisos, la inecuación se transforma en una igualdad dado que el cuantificador denotaun valor preciso; si son borrosos o imprecisos, cada una de las inecuaciones denotará loslímites superiores e inferiores de los cuantificadores, definidos como intervalos o conjuntosborrosos (por ejemplo, aproximación mediante núcleo-soporte). Esto nos permitió identificarunívocamente regiones del universo del discurso del silogismo definidos por cada una de laspremisas.

En el paso iii) se selecciona el método de optimización adecuado para cada caso. En estainvestigación usamos dos: Simplex, para los casos en los que el cuantificador de la conclu-sión es absoluto y programación fraccional para los casos en los que el cuantificador de laconclusión es proporcional. Para la correcta aplicación de dichos métodos es necesario añadirtres restricciones adicionales: i) que no hay ningún conjunto con un número negativo de ele-mentos; ii) en las premisas con forma fraccional, no hay ningún denominador igual a 0; iii) lasuma de las cardinalidades de todos los subconjuntos es igual a la cardinalidad del referencial.

Evaluamos este modelo en el tratamiento del silogismo aristotélico y de los otros modelossilogísticos analizados, obteniendo unos resultados consistentes. Además, también obtuvimosotras mejoras significativas, como el uso de combinaciones booleanas de términos, el tra-


tamiento de diferentes tipos de cuantificadores en un mismo argumento y el modelado deargumentos con cualquier número premisas o términos. Por otra parte, al adoptar una aproxi-mación heurística, abrimos la posibilidad de usar el silogismo en contextos no previamenteconsiderados. También cabe destacar que este modelo ha sido parcialmente implementadoen la librería software SEREA (Syllogistic Epistemic REAsoner) con la que se realizaron lamayor parte de los ejemplos.

La segunda parte del objetivo ii) fue el tratamiento de los términos vagos en el encadena-miento. Esto es especialmente significativo, puesto que es el nexo entre los términos extremos.Aquellos silogismos con un encadenamiento no exacto son considerados falaces (falacia de

los cuatro términos) por la lógica clásica, aunque sean aceptables desde el punto de vista delrazonamiento ordinario. Este tipo de casos se da, por ejemplo, cuando el término medio estádesempeñado por dos palabras de significado similar.

En la lógica borrosa existen diferentes aproximaciones al tratamiento de la similitud entreconceptos (como la inclusión borrosa o el solapamiento borroso), pero nosotros la aborda-mos usando la sinonimia, que es la relación de similitud lingüística por excelencia. Para eldesarrollo de nuestra propuesta [55] asumimos tres principios: i) la sinonimia es una relaciónaproximada que refleja un parecido de familia entre términos; ii) dos términos son sinónimossi aparecen como tales en un diccionario de sinónimos (thesaurus; como WordNet en nuestrocaso); iii) utilizamos diversas medidas de distancia entre conjuntos para medir o establecer elgrado de sinonimia entre dos palabras.

Cada palabra incluida en WordNet está descrita por una serie de glosas (breves descrip-ciones conceptuales de cada una de las acepciones del término), y cada una de ellas tieneasociado un conjunto de sinónimos (synset). Si dos términos son sinónimos es plausible pen-sar que sus synsets compartirán un elevado número de sinónimos, por lo que asumimos estarelación como principio. Para obtener un valor cuantitativo de esta medida, aplicamos algunade las métricas de distancia (coeficiente de Jaccard, de Solape,. . . ) sobre los synsets de lostérminos en comparación: si el resultado es 1, los términos son idénticos; si es 0 los términosno se parecen en nada.

Sin embargo, las distintas medidas generan resultados significativamente diferentes, porlo que surge la cuestión de cuál usar en cada caso. Para abordar este problema, proponemosusar las glosas. Si los términos en comparación comparten muchas de ellas, deben utilizarsemétricas que ofrezcan unos valores más elevados, como el coeficiente de Solape; si compartenpocas, se usarán las de resultado más conservador, como el coeficiente de Jaccard.


Respecto del patrón de inferencia, apuntamos que el encadenamiento aproximado tieneque verse reflejado en la fiabilidad del argumento. En este caso proponemos cualificar la con-clusión del argumento mediante un grado de confianza bajo el siguiente principio: cuantomayor sea el grado de sinonimia entre los términos, mayor es el grado de confianza en laconclusión. Así pues, la conclusión de un silogismo con un grado de similitud alto en el enca-denamiento aproximado será más fiable que uno con uno bajo. El desarrollo de este modelotambién es una contribución efectiva a la mejora de la expresividad del silogismo, pues nospermite tratar un tipo de inferencia aproximada usando un concepto típicamente lingüísticocomo la sinonimia.

Nuevos casos de uso del silogismo

Como indicamos, nuestro modelo silogístico tiene una mayor capacidad expresiva, que nospermite considerar usos del silogismo no contemplados hasta ahora. En particular, abordamosel caso del razonamiento probabilístico, dado que es un tipo de formalismo ampliamente estu-diado y utilizado en la Inteligencia Artificial y dada la equivalencia entre las oraciones cuanti-ficadas proporcionales y las probabilidades condicionales. Analizamos tres modelos distintos:el Teorema de Bayes Generalizado, las redes bayesianas y el razonamiento probabilístico enel ámbito judicial.

El Teorema de Bayes Generalizado se caracteriza por usar exclusivamente probabilidadescondicionadas. Para razonar con él, se necesita de una combinación entre el Patrón I de la In-

terval Syllogistics y el propio teorema y aplicar una serie de pasos iterativos para obtener losresultados óptimos. Usando nuestro modelo de silogismo mostramos como construir un argu-mento silogístico en el que se incluyen todas las probabilidades condicionadas y así obteneren un único paso la mayoría de los resultados óptimos.

Las redes bayesianas son ampliamente utilizadas en el modelado de situaciones con incer-tidumbre. Destacan por su gran versatilidad y su capacidad para ejecutar diferentes tipos deinferencias. A partir de un ejemplo de un diagnóstico médico, se ilustra como elaborar un si-logismo equivalente y obtener los mismos resultados en la ejecución de las inferencias básicasde las redes bayesianas (predictiva simple, predictiva con evidencia, diagnóstico e intercau-sal). Un silogismo, basado en oraciones cuantificadas del lenguaje natural y una estructurade premisas-conclusión, resulta más sencillo para usuarios no familiarizados, dado que lasredes bayesianas son un formalismo que usualmente requiere la intervención de expertos. Por


otra parte, el silogismo también nos permite incluir vaguedad o imprecisión con sencillez,simplemente substituyendo los cuantificadores precisos por cuantificadores vagos.

Por último, analizamos el uso del razonamiento probabilístico en la argumentación jurídi-ca. En ciertas ocasiones, esta no se aplica correctamente y aparece la llamada falacia del fiscal,que consiste en un uso erróneo del razonamiento probabilístico con el objetivo de probar laculpabilidad de un acusado. Usando nuestro modelo, todas las probabilidades pueden ser ex-presadas en un único argumento mediante oraciones cuantificadas y, dado que el cuantificadorde la conclusión busca el intervalo de máxima compatibilidad, si el conjunto de premisas noestá adecuadamente definido se obtendrá una indeterminación. Así pues, la expresión de es-te tipo de argumentos usando nuestro modelo facilitaría la identificación de argumentos malfundamentados.

Para concluir, consideramos que hemos desarrollado satisfactoriamente todos los aspectosrelevantes de nuestra hipótesis de trabajo. El modelo desarrollado presenta una capacidad ex-presiva significativa, que permite el tratamiento de argumentaciones complejas que incluyentanto cuantificadores borrosos como un encadenamiento aproximado. Esto, conjuntamentecon la perspectiva heurística adoptada, permite abrir nuevas posibilidades de uso para el silo-gismo, como la expresión de las redes bayesianas usando oraciones cuantificadas.

Como trabajo futuro apuntamos la incorporación de otros cuantificadores de la Teoría delos Cuantificadores Generalizados, como los ternarios o cuaternarios; el tratamiento de redesbayesianas más complejas (más allá de los casos básicos); explorar otros conceptos para eva-luar la similitud entre términos, como la analogía o la antonimia y el modelado del contexto.Por último, también proponemos avanzar en el desarrollo de SEREA, desde su versión actual(librería, que también debe ser mejorada), a una herramienta software gráfica más completa yanáloga a las existentes para el modelado de las redes bayesianas.

Preface

Reasoning takes part in any human activity and, by this omnipresence, its study has been aconstant in human thinking and in our understanding about the world. We reason using infer-ences, which are expressed in natural language using arguments, and they have the followingstandard structure: i) premises, which are a set of statements; ii) conclusion, a statement in-ferred from these premises. However, preserving this structure, the validity of the inferencescan vary significantly depending on the context or on its aim. For instance, let us consider thecase of B. Russell’s Inductivist Turkey [70]:

The turkey found that, on his first morning at the turkey farm, he was fed at 9a.m. Being a good inductivist turkey he did not jump to conclusions. He waiteduntil he collected a large number of observations that he was fed at 9 a.m. andmade these observations under a wide range of circumstances, on Wednesdays,on Thursdays, on cold days, on warm days. Each day he added another observa-tion statement to his list. Finally he was satisfied that he had collected a numberof observation statements to inductively infer that “I am always fed at 9 a.m.”.However on the morning of Christmas eve he was not fed but instead had histhroat cut.

Inductive arguments allow us to make a prediction in the future according to the informa-tion collected in the past about the same event. In this case, the premises are all evidence thatthe turkey gathered during the days in the farm and the (wrong) conclusion was “I am alwaysfed at 9 a.m.”. Let us now consider the following example of inference [46]:

All the beans from this bag are white. These beans are white. Therefore, thesebeans are from this bag.

10 PREFACE

In this case, the premises are “all the beans from this bag are white” and “these beans arewhite” and the conclusion “these beans are from this bag”. This is not an inductive argumentsince there is not a collection of evidence that allows us to infer something about the futurebut the conclusion is rather a hypothesis which explains the premises. This is the so-calledabductive reasoning, and, as we can see, with the same structure of premises and conclusionit generates a different type of argument.

The type of reasoning par excellence is perhaps deductive reasoning. It is characterized bybeing sure, definitive and free of controversy [65], which gives us a solid base for supportingour knowledge. For this reason, it is the dominant one in the scientific field, particularly inprobative fields such as mathematics.

The first systematic study of deductive reasoning in the history of logic was developed byAristotle in Ancient Greece [4]. He focused on the so-called syllogism. This is a deductiveargument of a very specific sort: it only deals with statements having the form All S are P

or Some S are P (and their respective the negated versions), where S stands for the subjectand P stands for the predicate and with arguments comprising three terms or concepts, twopremises and one conclusion. It was the hegemonic logical system from Ancient Greece untilthe nineteenth century, when mathematical logic appears [29]. At this moment, AristotelianSyllogistics was completed rejected. In [70, p. 202], B. Russell says:

I conclude that the Aristotelian doctrines with which we have been concernedin this chapter are wholly false, with the exception of the formal theory of thesyllogism, which is unimportant. Any person in the present day who wishes tolearn logic will be wasting his time if he reads Aristotle or any of his disciples.None the less, Aristotle’s logical writings show great ability, and would have beenuseful to mankind if they had appeared at a time when intellectual originality wasstill active. Unfortunately, they appeared at the very end of the creative periodof Greek thought, and therefore came to be accepted as authoritative. By thetime that logical originality revived, a reign of two thousand years had madeAristotle very difficult to dethrone. Throughout modern times, practically everyadvance in science, in logic, or in philosophy has had to be made in the teeth ofthe opposition from Aristotle’s disciples.

Nevertheless, in the second half of the twentieth century new perspectives about Aris-totelian Syllogistics appear [33, 13] and it was partially recovered. It is relevant to note thatall syllogistic statements are quantified statements and, quantification is a typical phenomenon

PREFACE 11

of natural language. In [86], D. Westersthål shows the compatibility between Aristotelian Syl-logistics and the Theory of Generalized Quantifiers (TGQ), the most complete and up-to-datetheory of quantification in linguistics, opening thus the possibility of interpreting syllogismas a form of deductive reasoning about quantities.

Nonetheless, quantification not only denotes precise quantities but also vague ones. Forinstance, terms such as “most”, “many”, “few”, etc. are quantifiers that are very commonin natural language and refer to a more or less indeterminate quantity, but are usually suffi-ciently informative to the individuals, even for making inferences. However, as stated above,Aristotelian Syllogistics only dealt with two types of propositions (which comprise the uni-versal quantifier “all” and the existential one “some” with their corresponding oppositions);therefore, a relevant field for the development and expansion of syllogistics beyond Aristotle’sscope with the combination of deductive reasoning involving vague quantities appears.

There are several different proposals that expand syllogistic reasoning following the Aris-totelian concept of quantified statement; i.e., a quantifier denotes a quantity relationship be-tween two sets. Some of these proposals are based on an extension or generalization ofthe Logic Square of Opposition (LSO), the Aristotelian classification of syllogistic quanti-fied statements, such as Intermediate Syllogistics [63] of Generalized Intermediate Syllogis-tics [37, 36]; others are based on a reinterpretation of quantifiers in terms of precise quantities,such as the Exception Syllogistics [38, 39]; others on the use of intervals to represent vaguequantifiers, such us the model developed by D. Dubois et al. [17, 18, 16]; and, another on theinterpretation of vague quantifiers through fuzzy sets such us Zadeh’s proposal [92].

However, during the twentieth century alternative interpretations for the quantifiers alsoappear, where a quantifier can also denote probabilities, beliefs or even modal references.The most used one is the interpretation of quantified statements in terms of conditional prob-abilities, as it opens up the possibility of a combination between syllogism and Bayesianreasoning [3, 16], a type of probabilistic reasoning that has been deeply studied. In this case,it is assumed that any proportional quantifier, such as “all”, “most”, “few”, etc. denotes prob-ability. Thus, for instance, “all mammals are animals” is equivalent to saying “if somethingis a mammal then it is an animal with 100% probability”, in the case of vague quantifiers,such as “most mammals live on land” is equivalent, for instance, to saying “if something is amammal then it lives on land with more than 80% probability” [43].

Quantifiers can also be interpreted as beliefs. In [79], M. Spies proposes to use Dempster-Shafer’s Theory, where the belief is represented through an interval. Thus, a quantifier denotes

12 PREFACE

the belief of a person in the relationship between the terms involved in the statement. Forinstance, “all mammals are animals” means that “I believe that if something is a mammal thenit is an animal with [1,1] degree”, in the case of vague quantifiers, such as “most mammals liveon land” means “I believe that if something is a mammal then it is an animal with [0.8,0.9]degree”. Starting from this concept, M. Spies [79] built a syllogistic framework that give us anew perspective about this kind of reasoning.

Other approach, elaborated by D. Schwartz [71], proposes interpreting quantifiers in termsof modality. For instance, “most mammals live on land” means, for instance, “usually, mam-mals live on land”, where the modifier “usually” assumes the meaning of the quantifier“most”. However, the treatment of inference patterns, as we shall explain, does not followthe standard characteristics of syllogistic reasoning but is based on Modus Ponens. Despitethis, it constitutes an alternative approach to syllogistic reasoning to be considered.

All the previous approaches, irrespective of how they model the meaning of a quantifiedstatement, only deal with a very limited number of inference patterns. For instance, Inter-mediate Syllogistics deals with more than one hundred inference patterns, but all of them aregrouped into the four Aristotelian figures. Fuzzy proposals and those with an alternative inter-pretation of quantified statements also manage a reduced number of patterns, although someof them are different from Aristotle’s ones. Thus, although the capability for managing vaguestatements of natural language was notably increased, they were only focused on a particulartype of quantifiers (mainly on proportional ones), disregarding the other ones considered bythe TGQ, and on a very reduced number of patterns of inference with no more than three orfour premises, thus significantly limiting the capability of the syllogism to manage moderatelycomplex problems.

On the other hand, vagueness can also appear in the terms of the syllogism, not only inquantifiers. In this case, the syllogism’s probative character is replaced by other approximatedor probable one since the links between vague terms are based on similarity rather than equal-ity. Thus, we here find another way of extending syllogistic reasoning. We think in additionthat, since we shall approach syllogism from the perspective of natural language, the conceptof synonymy is the linguistic relationship that best represents semantic similarity between twoterms. There are currently tools available on line for obtaining synonym relationships betweenterms, such us WordNet [1] or Galnet [26], which offer us the environment for proposing away of establishing the synonymy between two terms using measures of similarity.

PREFACE 13

Thus, taking into account all the ideas explained above, we propose in this research ex-tending syllogistic model of reasoning dealing with vagueness into two ways: managing vaguequantifiers and vague terms. In the first case, we use the framework of the TGQ to model themeaning of the quantifiers statements; Venn diagrams to build the knowledge representationand algebraic methods to build the reasoning process. In the second case, we use the conceptof synonymy, taken from linguistics and the philosophy of language, to measure the similaritybetween two terms; distance measures to calculate the similarity between two concepts andfuzzy inferences rules to develop the reasoning procedure associated to this kind of arguments.As we can see, this research constitute a multidisciplinary work where several fields such uslogic, philosophy of language, linguistics, pragmatics and computer science are involved todevelop a model of reasoning for dealing with arguments expressed in natural language thatcomprise fuzzy quantified statements.

Hypothesis

The current models for syllogistic reasoning only deal with a small part of syllogisms thathuman beings can manage and exhibit certain limitations, in the quantifiers, terms andinferences schemas that they manage. This makes them hardly applicable to realisticscenarios related to everyday reasoning. The existing proposals only consider argumentsof a very specific type many of them differing greatly from the use that people do in their dailylife. The development of a new model for syllogistic reasoning compatible with the TGQ anddealing with vagueness would contribute to improve the capability of Computing with Wordsto manage natural language arguments.

Objectives

Thus, the main aim of this research is to define a model for dealing with syllogistic argu-ments that is able to manage the quantifiers defined by TGQ and approximate terms tomake inferences expressed in natural language. The achievement of this aim is supportedby the following objectives:

1. Studying the existing proposals about syllogistic reasoning to identify how quanti-fied statements are interpreted. The analysis of syllogistic reasoning was framed inthe discipline of logic from the very beginning and the analysis of quantified statements

14 PREFACE

has changed gradually over time. The objective of this step is to analyse some of theseproposals and, after an in-depth analysis, to identify which of them are the most ade-quate to our aim. We also adopt Aristotelian Syllogistics as the referential framework,since it is the classical system.

2. Defining a model for the representation of quantified statements from the pointof view of reasoning and the TGQ. We assume the Sapir-Whorf hypothesis [28] and,hence, depending on the structure assigned to quantified statements, we fix a particularconceptualization of the world. Thus, in this step, we define the knowledge representa-tion associated to quantified statements which also determines the general form of ourinference procedure.

3. Elaborating a procedure to infer a valid conclusion that manages different quan-tifiers simultaneously. Attending to the two previous objectives, we define the stepsthat constitute the reasoning procedure and how it operates to infer a valid conclusionfrom a set of premises.

4. Studying different alternatives to measure the similarity between two terms in ap-proximate syllogisms. The validity of syllogisms with an approximate chaining di-rectly depends on how similar the involved terms are. Hence, in this stage we addressdifferent proposals that analyse this concept.

5. Proposing a mechanism to execute inferences with approximate chaining in themiddle term. Incorporating vagueness or approximation in middle terms entails a re-configuration of syllogistic inference as it focuses principally on the deduction of thequantifier. We provide an approach for measuring the similarity between two conceptsand how its results are interpreted in terms of validity or reliability of the syllogisminstead of in terms of quantifiers, as in the previous problem.

6. Analysing the compatibility between some types of Bayesian reasoning and syllo-gism supported in the equivalence between quantified statements and conditionalprobabilities. It is accepted that quantified statements can be interpreted in terms ofconditional probabilities. Thus, if we extend this equivalence to reasoning, certain typesof Bayesian reasoning can also be modelled using syllogisms. We shall explore thispossibility and the limitations that it can have and we shall illustrate how our proposal

PREFACE 15

works with some examples taken from real scenarios; particularly, medical diagnosisand legal argumentation.

7. Implementing our proposal of syllogistic system in a software library for testingit and making it available for any possible user. This library allows us to check thebehaviour of our model and facilitates its use by any user.

Structure of the thesis

To adequately develop the formulation of our problem and achieve the explained objectives,this thesis is organized as follows:

– Chapter 1: We introduce syllogistic reasoning as a model of deductive reasoning basedon the use of quantified statements. First of all, we explore the different interpretationsabout them described in the literature as quantified statements are the propositions man-aged in this type of reasoning. They are classified into two groups: i) those where theterms designate sets and the quantifier denotes a quantity link between them; and ii)those where the terms constitute the antecedent and consequent of a conditional propo-sition and the quantifier denotes the strength of the link between them. In the firstcase, we describe the main characteristics of Aristotelian Syllogistics as the first syl-logistic system and assume it as the set of inferences that any further proposal mustsatisfy. Thus, the collection of syllogistic frameworks described in this chapter arecomplemented with a critical analysis to check their compatibility with the Aristoteliansystem. In the second case, we describe three possible models for the conditional inter-pretation; i.e., as probabilities, as beliefs or as modality. We conclude the chapter withan analysis about which is the best model for satisfying our objective.

– Chapter 2: We present the definition of quantifiers according to TGQ and develop thefirst part of the main objective of this research; i.e., a proposal of syllogistic reasoningthat can manage the different quantifiers proposed by TGQ simultaneously. We includesome illustrative examples to show how it works and conclude by summarizing its maincharacteristics, and its strengths and weaknesses.

– Chapter 3: The question of approximate middle terms is proposed, including also abrief description of how this problem is addressed in the literature. The first part is abrief description of different environments and strategies for measuring the similarity

16 PREFACE

between two terms. We choose synonymy as the most interesting relationship sinceit is characteristic of natural language. We adopt a definition based on the use of athesaurus, a dictionary of synonyms, and different mathematical measures of similarity.Over these ideas, we propose an inference pattern, compatible with the one devoted toquantifiers, which allows us to obtain qualified syllogisms according to their reliabilityor confidence. We also include some illustrative examples and conclude by summingup our main contributions.

– Chapter 4: We study the compatibility between our model of syllogism with differenttypes of probabilistic reasoning. We analyse the so-called Generalized Bayes’ Theo-rem, Bayesian networks probabilistic reasoning in legal argumentation. We focus prin-cipally on the basic Bayesian inference patterns and how they can be modelled usingsyllogisms, providing some guidelines. We illustrate the behaviour of our method withan example taken from medicine and other from law. The objective is to show the pos-sibilities of syllogism as an alternative tool, easier and clearer for non-specialized users,to model some probabilistic problems that involve Bayesian reasoning.

– Conclusions: This is devoted to explaining the conclusions reached during this re-search. We summarize the main contributions made during this work, emphasizing itsstrengths and weaknesses, and also the most relevant open questions that we considerwill constitute our future work.

Publications

By way of conclusion, we would like to emphasize that the main results achieved from thisresearch have been described in several papers:

– The analysis about the behaviour of fuzzy approaches involving fuzzy quantified state-ments to deal with Aristotelian Syllogistics, the classical system of syllogism, was pub-lished in Fuzzy Sets and Systems [51] concluding that any of them could manage itadequately. A number of initial results were presented at at IEEE World Congress on

Computational Intelligence (WCCI2010) [50]. Both papers are our starting point to ex-plore the possibility of other models of syllogistic reasoning that involves both systems.

– Our proposal of syllogistic reasoning was explained in detail in Fuzzy Sets and Sys-

tems [51] and constitutes main heart of the Chapter 2. A number of preliminaries re-

PREFACE 17

sults have been previously presented at the XVI Congreso Español sobre Tecnologías y

Lógica Fuzzy (ESTYLF 2012) [57].

– The first approach to a syllogism involving approximate middle terms was presented atthe Congreso Español de Lógica, Metodología y Filosofía de la Ciencia en España [54]and subsequently extended in the paper presented at the 2013 IEEE International Con-

ference on Fuzzy Systems [55]. Both papers constitute the basis of the proposal de-veloped in chapter 3. A new paper comparing and analysing different procedures formeasuring synonymy was submitted to the XVII Congreso Español sobre Tecnologías

y Lógica Fuzzy (ESTYLF 2014) [53].

– We propose the field of argumentation theory as an adequate environment for the use ofsyllogistic arguments as it deals with any human situation that involves the intentionalacts of persuading or dissuading somebody. The first contact this field was throughpragmatics, the field that deals with utterances, to analyse the most superficial layer ofarguments. This analysis allowed us to collaborate with the research unit of Cognitive

Computing: Computing with Perceptions of the European Centre for Soft Computing

writing two papers, [52] and [19].

– In chapter 4, a brief section about the relevance of implicit premises in arguments isincluded. The links between the concept of intentionality and the use of arguments inan approximated framework were initially considered in [48]. An extended version ofthis paper was published in Mathware & Soft Computing journal [49].

– The first steps of this research also led us to make a number of reflections on the impactand the role of vagueness in fuzzy logic from the point of view of philosophy. Theywere presented in the 2009 IFSA World Congress - 2009 EUSFLAT Conference (IFSA-EUSFLAT 2009) [49] and contributed to define the perspective adopted in this research.

CHAPTER 1

STATE OF THE ART IN SYLLOGISTIC

REASONING

Human beings, in our daily lives, have to deal with quantitative aspects of reality; that is, weneed to refer to the number or the amount of elements endowed with a particular feature orwhich fulfil a property. Natural language, a characteristic capability of zoon politikon [5]1,offers us a very useful tool for expressing quantities: linguistic quantifiers. The most commonare, perhaps, terms such as “no”, “many”, “few”, etc. that denote proportional values; otherslike “thirty”, “ten”, “five”, etc. for referring to absolute quantities; “all but five” or “all butthree” are useful for designating exceptions, focusing on the number of elements that do notfulfil the property; “double” or “triple” are relationships of multiplication, etc. The TGQ [8]identifies all these different types of quantifiers while classical logic only consider two: theuniversal (usually expressed as “all”) and the existential (usually expressed as “some”) ones.

Linguistic quantifiers can be used directly in simple statements (i.e., “How old are you? Iam twenty eight years old”); however, they are mostly used inside in propositions that involvea relationship between two terms or sets: the so-called binary quantified statements2. Forinstance, in order to tell somebody about how many students are attending a conference, onehas to refer to a quantity, and, hence, use a quantified statement such as “twenty-five studentsattended the conference” or “almost all students attended the conference”. Depending on the

1Although the usual translation of this expression is “political animal”, the characterization of the human beingas a “linguistic animal” is understood by it.

2The research presented in this thesis is mainly focused on binary quantifier statements and, for brevity, they willbe referred simply as quantified statements.

20 Chapter 1. State of the Art in Syllogistic Reasoning

available information, the quantities can be precisely known (as in the first statement) andreferred to using precise quantifiers such as “one hundred”, “twenty five”, “50%”, “no”, etc.;or they can be vague or imprecisely known (as in the second statement), using, in this case,fuzzy quantifiers such as “few”, “many”, “most”, etc.

Regarding the pragmatic dimension of quantified statements, it is relevant to note theirexpressive wealth for transmitting connotations and a palette of shades that are presenting inany event. For instance, considering again the previous example, instead of using the absolutequantifier “twenty-five” and knowing how many students are in the classroom (i.e.; twenty-eight), one could use “all but three students attended the conference”, thus emphasizing, likethis, that the conference caught the attention of almost all students. However, if somebodywants to emphasize that three students did not attend the conference, he could say “somestudents did not attend the conference” pointing out implicitly, in this way, that the conferencemay not have been interesting enough for the students.

On the other hand, quantified statements can also be used for making inferences whereall the involved statements are quantified; i.e., to deduce a conclusion (a quantified statement)from a set of premises (also quantified statements). The standard form of the quantified state-ments for this purpose is Q S are P, where Q stands for the quantifier (either precise or vague),S stands for the subject of the statement, P stands for its predicate and are stands for the cor-responding form of the copulative verb to be. For instance, the statements “most studentsattended the conference” or “twenty-five students attended the conference” follow this struc-ture. Let us now consider a very basic example of inference involving quantified statements;i.e., if we say “almost all students attended the conference” we can infer that “very few stu-dents did not attend the conference” since the quantifier “almost all” denotes a high quantity,while “very few” denotes the complementary quantity. The most common form of syllogismis composed by two premises and a conclusion and the engine of the inference is the chaining;i.e., it is possible to conclude that two terms (extreme terms) are linked because there is a thirdterm that links them: the so-called middle term. The usual name for this kind of inference issyllogism and the logical system generated by its study is known as syllogistics.

Thus, the aim of this chapter is to introduce the topic of syllogistic reasoning and reviewthe state of the art to provide a highly detailed characterization of the different models pro-posed, both in philosophical and fuzzy logic. We also assume Aristotelian Syllogistics asthe referential framework that any further extension of syllogism must include as a partic-ular case. We classify the syllogistic systems analysed into two groups according to their

1.1. Set-Based Interpretation 21

interpretations of quantified statements: i) set-based interpretation (section 1.1); and ii) con-ditional interpretation (section 1.2). Finally, section 1.3 presents an analysis comparing bothinterpretations.

1.1 Set-Based Interpretation

Set-Based interpretation assumes that the terms that constitute a quantified statement in itsstandard form (Q S are P) are sets. A set, adopting an intuitive definition [12], refers to acollection of objects that share a particular characteristic or property. Let us consider thestatement “all natural numbers are real numbers”; the term natural numbers denotes the setcomposed by natural numbers; the term real numbers denotes the set of real numbers and all

how the elements of both sets are linked. Hence, any quantified statement denotes a relation-ship of quantity between sets, the terms that comprise it being term-sets. Thus, the subject ofthe statement is called subject-term and its predicate, the predicate-term. Furthermore, theyconstitute precise assertions although they can involve vague quantities or a vague referenceto a quantity. Thus, the general rule for interpreting the schema Q S are P is that there is a

quantity-relation between S and P denoted by Q.This section describes different syllogistic systems that can be found in the literature and

it is organized as follows: section 1.1.1 describes Aristotelian Syllogistics, the first syllogisticsystem in the history of logic; section 1.1.2 analyses the so-called Intermediate Syllogistics,the first extension of Aristotelian Syllogistics; section 1.1.3 characterizes the named Excep-tion Syllogistics, which is based on the exclusive use of exception quantifiers; section 1.1.4describes a fuzzy syllogistics based on the use of intervals; section 1.1.5 details the maincharacteristics of another fuzzy syllogistics based on fuzzy numbers and fuzzy arithmetic;and, finally, section 1.1.6 presents a generalization of Intermediate Syllogistics in fuzzy logic.

1.1.1 Aristotelian Syllogistics

Aristotelian Syllogistics [4], developed in the fifth century B.C., is the first approach to syl-logistic reasoning and also the first logical system in history. Aristotle defined syllogism as adeductive argument “in which certain things having been supposed, something different fromthose supposed results of necessity because of their being so”[7, I.2, 24b18-20]. The things

supposed are the premises of the argument and results of necessity are the conclusion. It is


worth noting that syllogism has a probative and deductive character because talking about“results of necessity”, where the premises are true, prevents the conclusion from being false.

The systematic study of syllogism was developed by Aristotle in the books that comprisethe Organon [6]. For the analysis of syllogism, the most relevant books are De Interpretatione,where Aristotle’s theory of categorical statement is described, and Prior Analytics, whichincludes Aristotelian theory of deductive inference.

This section is organized as follows: section 1.1.1.1 addresses the theory of Aristoteliancategorical statements; section 1.1.1.2, describes how the classical version of the Logic Squareof Opposition is generated; section 1.1.1.3, describes the Aristotelian reasoning patterns and,finally, section 1.1.1.4 concludes with a summary analysing the strengths and weaknesses ofthis proposal.

1.1.1.1 Theory of Aristotelian Categorical Statements

The theory of categorical statement describes the so-called categorical statements, the typicalpropositions of syllogisms. These are defined as “assertions about classes3 that affirm or denythat a class is included in other total or partially” [12, p. 168]. As we can see, the concept ofset is the key point of this framework.

It is worth noting that Aristotle assumes that the logical structure and the syntactical struc-ture of a statement are the same and, hence, the logical terms (subject-term, predicate-termand copulative verb) coincide with the syntactical categories (subject, attribute and copulativeverb, respectively). At the beginning of the twentieth century, the new logic developed by G.Frege and B. Russell dismissed this idea. For them, the logic structure of propositions doesnot coincide with their syntactical structure but corresponds with the mathematical structureof function-argument [29]. It is also relevant to say that every categorical statement is quan-tified; i.e., it always involves some kind of quantity. Those statements that do not involvequantifiers but singular terms, such as proper names (i.e.; “Arthur Conan Doyle is a writer”),are not categorical statements and, thus, are not valid for making categorical syllogisms.

The properties of categorical statements are as follows [6]:

1. Quantity: It is universal if the statement refers to each of the elements that belong tothe subject-term and it is particular if the statement only refers to a portion of them.

3For the context of this research, class and set are equivalent terms.


Universal quantity is expressed by the quantifiers “all” and “no” and particular quantityby the quantifiers “some” and “some. . . not”.

2. Quality: It is affirmative if the assertion affirms a universal or a particular quantity andit is negative if it denies the quantity.

3. Distribution: It is a property of the statements respect to the terms that it involves. Astatement distributes a term “if it refers to each member of the class designated by thatterm” [12, p. 168]. As will be shown, this is a key concept for the development of thetheory of syllogism.

The combination of these properties generates the four standard categorical statements [6]:

– All S are P: This is a universal affirmative and means that every member of the subject-term is also a member of the predicate-term; i.e., “all natural numbers are real numbers”.

– No S is P: This is a universal negative and means that the subject-term and the predicate-term do not share any element; i.e., “no stone is a vegetable”.

– Some S are P: This is a particular affirmative and means that certain members of thesubject-term are also members of the predicate-term; i.e., “some animals are pets”.

– Some S are not P: This is a particular negative and means that certain members ofthe subject-term are not members of the predicate-term; i.e., “some animals are notmammals”.

Table 1.1 summarizes the four Aristotelian categorical statements. The third column fromthe left shows the capital letters commonly used for referring to them. This nomenclaturecomes from Scholastic philosophers in the Middle Ages and it has a mnemonic purpose, sincethe letters A and I stand for affirmative statements (universal and particular ones, respectively)and E and O for negative ones (universal and particular ones, respectively). A and I comefrom the Latin word, “AffIrmo”, that means “I affirm” and E and O come from, “nEgO”; thatmeans, “I deny”.

The property of distribution [12, 61] arises in each categorical statement from the proper-ties of quantity and quality according to the following description (see Table 1.2):

– All S are B: S is distributed and P is not distributed; i.e., in “all human beings aremortal”, human beings is distributed because we know that each of the elements in the


Type of proposition Form Capital letterUniversal affirmative All S are P AUniversal negative No S are P EParticular affirmative Some S are P IParticular negative Some S are not P O

Table 1.1: Four standard categorical statements.

Statement Subject-term Predicate-termA Distributed UndistributedE Distributed DistributedI Undistributed UndistributedO Undistributed Distributed

Table 1.2: Distribution in categorical statements.

set human beings belongs to the set of mortal beings; however, we do not have the sameinformation regarding the elements of the predicate-term.

– No S is B: S and P are distributed; i.e., “no stone is a vegetable” classifies every elementof both sets because no element in the set stones belongs to the set vegetables and viceversa.

– Some S is B: S and P are not distributed; i.e., “some animals are pets”, neither theelements of animals nor the elements of pets are classified.

– Some S is not B: S is not distributed and P is distributed; i.e., “some animals are notmammals” classifies every element of the set mammals as all of them belong to the setanimals but in this term-set there are unclassified elements.

1.1.1.2 Logic Square of Opposition

Aristotle’s Syllogistics is a binary logic, and therefore, only two truth-values are managed:truth and falseness. So, each one of the four categorical statements can be true or false, whichgenerates a determined set of dependences among them. These are the so called relationships

of opposition [12, p. 214-216]:

– Contradictory (A-O, E-I): Two propositions are contradictory if one is the denial ornegation of the other; whenever one is true, the other is false and vice versa. For


A: All S are P E: No S is P

I: Some S is P O: Some S are not P

Contraries

Subcontraries

Sub

alte

rnat

ions

Subalternations

Contradictories

Figure 1.1: Classical LSO.

instance, if “all mammals are animals” is true, then “some mammals are not animals”is false.

– Contrariety (A-E): Two propositions are contrary if they cannot both be simultane-ously true; i.e., the truth of either of them entails the falseness of the other; however,both can be false. For instance, “all animals are mammals” and “no animal is a mam-mal” are simultaneously false.

– Subcontrariety (I-O): Two propositions are subcontrary if they cannot both be false,although they may both be true. For instance, “some animals are mammals” and “someanimals are not mammals”.

– Subalternation (A-I, E-O): Two propositions with the same subject and predicate-terms and quality (affirmative or negative); the truthfulness of the the universal one-named superaltern- entails the truthfulness of the particular one -named subaltern.For instance, “no cat is a stone” entails “some cat is not a stone”.

In the Middle Ages, scholastic philosophers proposed a graphical representation of cate-gorical statements according to these properties. The result, called the Logic Square of Oppo-

sition (LSO), is shown in Figure 1.1.The LSO allows us to generate some simple inferences known as immediate inferences

as they are constituted by a single premise and two terms. They do not belong to the general


If A is true, then E and O are false, while I is true.If E is true, then A and I are false, while O is true.If I is true, then A and O are undetermined, while E is falseIf O is true, then A is false, while E and I are undetermined.If A is false, then O is true, while E and I are undetermined.If E is false, then I is true, while A and O are undetermined.If I is false, then A is false, while E and O are true.If O is false, then A and I are true, while E is false.

Table 1.3: Basic immediate inferences.

Convertend ConverseAll S is P Some P is S (by limitation)No S is P No P is SSome S is P Some P is SSome S is not P Conversion invalid

Table 1.4: Conversion inferences.

theory of syllogism but they play a fundamental role in its adequate development since they areused by Aristotle for proving the validity of his inference schemas. Two types of immediateinferences can be distinguished [12]: i) basic ones (see Table 1.3) and ii) derived ones, whichinvolve more than one property of the LSO. The latter are:

– Conversion (E,I,A): This allows us to obtain equivalent statements by interchangingtheir terms. E and I are simply convertible; i.e., subject-term and predicate-term can beinterchanged with no restriction. For instance, “no vegetable is a stone” is equivalent to“no stone is a vegetable”, as this quantity relationship is based on the intersection oper-ation, which is commutative. A statements are not simply convertible as the inclusionoperation is not commutative. However, in this case we can say that A statements areconvertible with restrictions because it is true that if “all animals are mammals” is true,then “some animals are mammals” is also true; i.e., the conversion of the subaltern. Fi-nally, the conversion of O statements is not possible as the statement and its convertedare completely independent. Table 1.4 summarizes this inference.

– Obversion: This allows us to obtain statements that are logically equivalent throughchanging only the quality of the statement and substituting the predicate-term by itscomplementary. Thus, for instance, the obverted form “no vegetable is a stone” is


Obverted ObverseAll S is P No S is non−P (by limitation)No S is P All S is non−PSome S is P Some S is not non−PSome S is not P Some S is non−P

Table 1.5: Obversion inferences.

Premiss ContrapositiveA: All S is P A: All non−P is non−SE: No S is P O: Some non−P is not non−S(by limitation)I: Some S is P Contraposition is invalidO: Some S is not P O: Some non−P is not non−S

Table 1.6: Contraposition inferences.

“all vegetables are no-stones”. As we can see, the quality of the statement changes (itis affirmative instead of negative) and the predicate term (stone) is substituted by itscomplementary (no-stone). Table 1.5 summarizes this inference.

– Contraposition: This allows us to obtain statements that are logically equivalent byusing the complementary terms of the subject and the predicate-terms interchanging itspositions. Thus, for instance, the contrapositive of “all mammals are animals” is “allno-animals are no-vegetables”. Table 1.6 summarizes this inference.

This version of the LSO shows the so-called problem of existential import [58]. A state-ment has existential content if it affirms that the sets denoted by each one of the involved termshave at least one member; in other words, all the designated sets are not empty [12]. I and Ohave existential import because their assertion necessarily entails the existence of elements inthe involved sets. For instance, the truthfulness of “some animals are mammals” entails thatthere is, at least, one element in the set animals that is also a member of the set mammals. Letus consider now the case of A and E statements. For instance, “all mammals are animals” istrue because we know that every member of set mammals belongs to the set animals; now, ifwe say “all Martian mammals are animals” is this true or false? From a strict logical pointof view, it cannot be false because every member of the subject-term Martian mammals ( /0)belongs to the predicate-term animals ( /0 is subset of every set); hence, it is true. Somethingsimilar happens with E statements. Thus, if we accept this interpretation for the A and O


statements (not involving the existential import), the subalternation property is not valid giventhe existential meaning of I and O. For this reason, the most accepted interpretation withinthe Aristotelian LSO is that A and E statements have existential import.

The debate on existential import has been raging in Philosophy since Aristotle’s day andwas very important during the Middle Ages [29]. Nevertheless, the most accepted interpre-tation of this question today is that the existential import is not always assumed in ordinarylanguage and, sometimes, we reason without assuming it [29, 58].

An alternative version of the LSO, according to this last idea, is proposed by S. Pe-ters and D. Westersthål [58]. Figure 1.2 shows the so-called Modern LSO. The quantifier“some. . . not” is substituted by “not all” and the relationships expressed in the LSO are mod-ified because the concept of ‘opposition’ is substituted by the concept ‘negation’ and, hence,the concept of truth ceases to be the core of the LSO. Negation can adopt three forms:

Outer negation This is the equivalent of the contradictory property in the Aristotelian LSO.The outer negation of Q S are P is not (Q S are P); for instance, the outer negation of“all mammals are animals” is “not all mammals are animals”. In the Modern LSO, theouter negation is in the diagonals.

Inner negation This is the equivalent of the contrariety and subcontrariety properties of theclassical LSO. Hence, the inner negation of Q S are P is Q S are not P; for instance, theinner negation of “some animals are mammals” is “some animals are not mammals”.In the Modern LSO, the inner negation is in the horizontal line.

Dual This is the combination of the outer and inner negation and it is the equivalent to theobversion property. Hence, dual of Q S are P is not (Q S are not P); for instance, thedual of “no vegetable is a stone” is “not all vegetables are stones”. It is worth notingthat a sentence and its dual are logically equivalent; i.e., if one of them is true or falsethe other is also true or false.

1.1.1.3 Theory of Aristotelian Syllogism

As we have already stated, the basic inference schema of Aristotelian syllogistics is the syllo-

gism. It is defined as a piece of deductive reasoning in which two general terms are linked bya middle term [7, I, 23 (40b16)] and requires three terms, two premises and a conclusion. This


All S are P No S is P

Some S is P Not all S are P

Inner Negation

Inner Negation

Dual DualOuter Negation

Figure 1.2: Modern LSO.

(a) Aristotelian syllogistic inference.

All human beings are mortalAll Greeks are human beings

All Greeks are mortal

(b) Inference schema.

MP Q1 DT are MTNP Q2 NT are DTC Q3 NT are MT

Table 1.7: Example of Aristotelian syllogism.

defines the syllogism as a form of mediate inference, in opposition to immediate inferencesexplained above, which only included two terms and a single premise.

The term shared by the premises is the so-called Middle Term (DT) and the other twoare the so-called Extremes, where the subject-term of the conclusion is the Minor Term (NT)and the predicate-term is the Major Term (MT). The most famous example of Aristoteliansyllogism is shown in Table 1.7(a). This example follows the inference schema shown inTable 1.7(b), where Q1, Q2 and Q3 are crisp quantifiers (“all”,”some”, “no” or “some. . . not”).The Major Premise (MP) is the premise in which the MT appears and the Minor Premise

(NP) in which the NT appears.As we have previously mentioned, the DT only appears in the premises although it can

be the subject-term or the predicate-term. For instance, in the schema of Table 1.7(b) it isthe subject-term in MP and the predicate-term in NP. The different positions of DT in thepremises generates the so-called four Aristotelian Figures (see Table 1.8).


Figure I Figure II Figure III Figure IVQ1 DT are MT Q1 MT are DT Q1 DT are MT Q1 MT are DTQ2 NT are DT Q2 NT are DT Q2 DT are NT Q2 DT are NT

——————– ——————– ——————– ——————–Q3 NT are MT Q3 NT are MT Q3 NT are MT Q3 NT are MT

Table 1.8: Figures of Aristotelian Syllogistics.

Figure I Figure II Figure III Figure IVAAA (Barbara) EAE (Cesare) AAI (Darapti) AAI (Bramantip)AIE (Celarent) AEE (Camestres) IAI (Disamis) AEE (Camenes)

AII (Darii) EIO (Festino) AII (Datisi) IAI (Dimaris)EIO (Ferio) AOO (Baroco) EAO (Felapton) EAO (Fesapo)

AAI (Barbari) EAO (Cesare) OAO (Bocardo) EIO (Fresison)EAO (Celaront) AEO (Camestrop) EIO (Ferison) AEO (Camenos)

Table 1.9: Moods of Aristotelian Syllogistics.

(MP) No vegetable is a stone(NP) Some vegetables are green(C) Some stones are not green

Table 1.10: Ferison syllogism.

Since there are four Figures and four possibilities for each one of the three premises (thefour categorical statements of the LSO), there are 4× 43 = 256 possible combinations ofinference patterns. Nevertheless, only 24 are valid: the so-called Moods (see Table 1.9). TheMoods are labelled with three letters, the first capital letter refers to the MP, the second oneto the NP and the last one to the C. As an example, syllogism of the Table 1.7(a) is of thetype AAA (Barbara) and it belongs to Figure I. Another example is Ferison (EIO) Mood (seeTable 1.10), which belongs to Figure III (the DT is the subject-term in both premises) and itsMP is an A statement and the NP and C are I statements.

Aristotle handles different ways for testing the validity of a syllogism: proofs, counterex-amples and rules.

Regarding the proofs, Aristotle distinguishes between perfect proofs and imperfect proofs.Perfect proofs, or perfect syllogisms, are in deductions that “need no external term in orderto show the necessary result” [7, 24b23-24]; imperfect proofs, or imperfect syllogisms, arein deductions that “need one or several in addition that are necessary because of the terms


(a) Invalid syllogism.

All roses are flowersSome flowers are redSome roses are red

(b) Counterexample.

All human beings are mortalSome mortal beings are dogsSome human beings are dogs

Table 1.11: Aristotelian counterexample.

supposed but were not assumed through premises” [7, 24b24-25]. In [33], J. Łukasiewicz,adopting a contemporary point of view, considers that those syllogisms that only need perfectproofs are the primitive rules or axioms of the system, while the others are derived from them.

Regarding the use of counterexamples, it is a well known procedure in first-order logic.The procedure is simple: following the same inference schema for the syllogism which isunder evaluation, trying to build an example with true premises and a false conclusion. Ta-ble 1.11 shows an example.

The last of the Aristotelian ways of determining the validity of a syllogism is to checkwhether it violates any of the rules of the syllogism. If all of them are satisfied, the syllogismis valid. The syllogistic rules are the following [7]:

– Distribution

R1 The middle term is distributed as least once.

R2 No term is distributed in the conclusion unless it is distributed in one premise.

– Quality

R3 At least one premise is affirmative.

R4 The conclusion is negative if and only if one of the premises is negative.

– Quantity

R5 At least one premise is universal.

R6 If either premise is particular, the conclusion is particular.

Apart of the Aristotelian ways of proving the validity of syllogisms, in the nineteenthcentury J. Venn [85] developed a graphical procedure, known as Venn Diagrams, for repre-senting categorical statements and categorical syllogisms. It was subsequently improved by


A B

A ∩ BA ∩ B A ∩ B

A ∩ B

Figure 1.3: Primary diagram of the term-sets A and B.

A B

x

Figure 1.4: Venn-Peirce diagram for “All A are B and Some B are not A”.

C. S. Peirce [47]. Its main concept is the primary diagram, which represents all the possibleset-theoretic relationships between the sets involved in a categorical statement identifying thefour possible quantities referred to by the quantifiers; i.e., A∩B, A∩ B, A∩B and A∩ B (seeFigure 1.3). Only two cardinalities can be represented: empty sets, denoted by shadow areas,and sets with at least one element, denoted by ‘x’. Figure 1.4 shows the representation for “allA are B and Some B are not A”.

Adding an additional set intersected with the standard two, categorical syllogisms also canbe represented. For instance, Figure 1.5 shows the Ferio (EIO) Mood of Figure I (see 1.5(a))and its corresponding representation using Venn-Peirce diagrams (see 1.5(b)).

1.1.1.4 Summary

Aristotle focused on a type of deductive reasoning, syllogism, which deals with a very spe-cific class of propositions, the so-called categorical statements, which are quantified. Hedistinguishes between four types according to the properties of quantity (universal or particu-lar), quality (affirmative or negative) and distribution (distributed and not distributed). These


(a) Ferio Syllogism.

(MP) No DT is MT(NP) Some NT is DT(C) Some NT is not MT

(b) Diagram.

MT DT

NT

x

Figure 1.5: Ferio syllogism and its representation using Venn-Peirce diagrams.

properties generate a set of dependences among them using the concept of opposition thatconstitute the LSO, one of the principal contributions of Aristotelian Syllogistics to the studyof natural language and quantification, and the basis for his theory of deductive arguments:syllogisms.

Syllogisms are arguments which comprise three terms, two premises and a conclusion.Their validity is supported by the chaining of the extreme terms, which appear in the conclu-sion, through a middle term, that only appears in the premises. Attending to these constraints,twenty-four valid inferences schemas are obtained: the so-called Moods.

Aristotle only dealt with a very restricted set of quantifiers and a very specific kind of argu-ments. Hence, the capability of this approach for managing many of the arguments expressedin natural language is very limited as many of them involve other quantifiers and vaguenessin its formulation. Nevertheless, Aristotelian studies establish the basis for the study of rea-soning involving quantities, preserving its natural language form, and we shall adopt it as thereferential framework for subsequently analysing syllogistic systems.

1.1.2 Intermediate Syllogistics

During the Middle Ages, one of the main topics of Scholastic Philosophy was AristotelianSyllogistics, making a number of relevant contributions, such as the LSO. However, it is inthe twentieth century when the first proposals for increasing the quantities dealt with in asyllogism appear: P. L. Peterson [60, 59, 61, 62, 63] and B. Thompson [81, 82] introducethe concept of Intermediate Quantifier, which extends syllogistic inferences beyond the fourAristotelian crisp quantifiers. In the following sections, we shall describe this concept, how itis built and the valid syllogisms that can be generated using them.


A: Most S are P E: Most S are not-P

I: Many S are P O: Many S are not-P

Contraries

Subcontraries

Sub

alte

rnat

ions

Subalternations

Contradictories

Figure 1.6: LSO with “most” and “many”.

This section is organized as follows: section 1.1.2.1 addresses the theory of intermediatecategorical statements and the new LSO generated; section 1.1.2.2 describes the reasoningpatterns and, finally, section 1.1.2.3 concludes with a summary analysing the strengths anddrawbacks of this proposal.

1.1.2.1 Theory of Intermediate Categorical Statements

The notion of intermediate categorical statement is an extension of the Aristotelian categor-ical statement incorporating the concept intermediate quantifier. It is defined as any quan-tity/quantifier between the universal (“all”, “no”) and the particular (“some”, “some. . . not”)quantifiers and consistent with the LSO [63]. This means that all the relationships definedin the Aristotelian LSO (contrary, subcontrary, contradictory and subalternation; see 1.1.1.2)are preserved. However, in natural language there a lot of quantifiers that can appear in thespace between the universal and particular quantities and, hence, there are multiple candidatesto be incorporated. In [60], three possibilities for the combination of the quantifiers “few”,“many” and “most” are considered : i) “few” and “many”; ii) “few” and “most”; iii) “most”and “many”. We shall focus only on option iii) because, as will be explained, it shows thebest behaviour.

Figure 1.6 shows the LSO built with “most” and “many” (Most-Many LSO), where allthe typical properties of a LSO can be observed. If, furthermore, the relationship shown in


Categorical statementsFew S are P Most S are not-PFew S are not-P Most S are P

Table 1.12: Equivalence between “few” and “most”.

Table 1.12 is also assumed, where the equivalence between categorical statements involving“few” and “most” is established, option ii) (a Most-Many LSO) and option i) (a Few-MostLSO) can be generated; therefore, the Most-Many LSO integrates the other two. This is aheuristic conclusion that was achieved after comparing the three generated LSO. In addition,the are other two reasons that make the Most-Many LSO preferable [60]:

– It emulates the Aristotelian LSO exactly: the affirmative statements are on the left-handside (A and I) and the negative ones are on the right (E and O).

– Most-Many LSO and Aristotelian LSO can be integrated into a single LSO, preservingall the properties and attending to the following order [60, p. 168]:

All S are P→Most S are P→Many S are P→ Some S are P (1.1)

No S is P→Most S are not-P→Many S are not-P→ Some S are not-P (1.2)

A LSO built with intermediate quantifiers, in addition to the existential import (see sec-tion 1.1.1.2; it is solved in the same way; i.e., subjects-terms cannot be empty sets), exhibitsother problematic questions: i) multiple quantities for the same quantifier [60, 81]; and ii) aconstant reference class [60].

Question i) arises owing to the vague and context-dependent character of intermediatequantifiers. Let us consider the following statement, “most students are dedicated”; a possibledefinition for “most” is that the number of students that are dedicated is greater than thenumber of students that are not dedicated. This definition opens a wide range of possibilitiesin the effective interpretation of “most” or any other intermediate quantifier. Thus, in [60, 81],three possible senses for delimiting this meaning are proposed:

– Minimal sense: This means at least or no less than the quantity named.

– Maximal sense: This means only or no more than.

– Exact sense: This is the combination of the minimal and the maximal senses; i.e., no

more nor less than.


The preservation of the subalternation property requires a particular combination of thesesenses. Thus, “some”, “many” and “most” are interpreted in the minimal sense and “few”in the maximal one, given that their minimal sense is too close to the usual interpretation for“some” [60, 81].

Question ii), the so-called constant reference to a class, is also characteristic of an LSOwith intermediate quantifiers. It supports the two aforementioned assumptions and refersto the possible readings of a quantifier; i.e., absolute or proportional. In the absolute one,the quantifier refers to the cardinality of the relationships involved in a proposition sayingthat it is at least n, where n is a quantity value. In the proportional reading, the quantifierrefers to a relative proportion of the sets involved in the corresponding proposition sayingat least k, where k is a fraction between 0 and 1 or a percentage [44]. Let us consider thefollowing example “many students are athletes” in the context of the University of Santiagode Compostela (USC) and in the academic year 2011/2012: 28,438. We can say that theproposition is true using the absolute reading if 4,000 or 5,000 students play sports regularly,as 4,000 or 5,000 students constitute a lot of people; however, if we adopt the proportionalreading, this quantity is around 15% and it is more difficult to accept that “many students areathletes” is true, given that the usual k value for “many” is over 50% [76]. A deep analysis ofthis question about the quantifier “many” can be found in [44, 21].

Coming back to the LSO, maintaining the same reference is fundamental for preservingthe properties as, otherwise, they are not fulfilled. For instance, attending to Most-Many LSO(see Figure 1.6 and Table 1.12), from most S are P we can infer few S are not-P wheneverwe maintain the proportional reading in both statements. If we use the proportional readingin “most” and the absolute one in “few”, then this inference is not valid. For instance, con-sidering again the students of the USC, if “most students are athletes” means “around 70% ofstudents are playing sports regularly”, there are around 8,000 of them that do not play sportsregurlaly, and, hence, we cannot say that 8,000 students are “few students”.

To increase the expressive capacity of the LSO, Peterson merges the Aristotelian LSO withthe Most-Many LSO and adds two additional quantifiers: “almost all” and “few”. This gen-erates a 5-quantity square [59, 63] as involves the five quantifiers “all”, “almost-all”, “most”,“many” and “some” preserving all the standard relationships (see Figure 1.7). The new in-


A: All S are P E: No S are P

I: Some S are P O: Some S are not-P

P: Almost-all S are P

K: Many S are P

T: Most S are P

B: Few S are P

D: Most S are not-P

G: Many S are not-P

Figure 1.7: 5-Quantity LSO.

termediate categorical statements are named according to the following labels, following themnemonic medieval spirit4:

– Predominant: almost-all S are P (P) and and few S are P (B)

– Majority: most S are P (T) and most S are not P (D)

– Common: many S are P (K) and many S are not P (G)

It is clear that the aforementioned quantifiers do not exhaust the space between the uni-versal and particular ones. Thus, given that there is a procedure for generating LSOs, Peter-son [63, p. 233] proposes to use fractions to generalize the concept of intermediate quantifier.In this way, any intermediate quantifier can be expressed by a fraction m/n, being n≥ m, andthe symbols >,≥,<,≤ for selecting the corresponding sense of the quantifier. For instance,> 2/3 the S are P denotes the intermediate quantifier “more than two thirds of S are P”. Thegeneral schema for any fractional quantifiers LSO is shown in Figure 1.8. It is worth notingthat only proportional readings of quantifiers can be used, and, therefore, any considerationof absolute ones lies outside this framework.

4P, T and K are consonant letters contrasting with the vowels of Aristotelian LSO. Furthermore, they arewidespread unvoiced consonantal stops which point of articulation moving from front to back. The letters for thenegative statements, B, D and G are their voiced counterparts.


More-than n-m/n S are P More-than n-m/n S are not-P

m/n S are P m/s S are not-P

Contraries

Subcontraries

Sub

alte

rnat

ions

Subalternations

Contradictories

Figure 1.8: General schema for generating the LSO for fractional quantifiers.

1.1.2.2 Theory of Intermediate Syllogisms

P. Peterson and R. Carnes build their doctrine of intermediate syllogism over the 5-quantityLSO (see Figure 1.7) [63, chap. 2]. Thus, any intermediate syllogism can be obtained by“taking any traditional syllogism (valid or not) and replace one or more of its categoricalpropositions with some intermediate categorical propositions” [63].

The syllogistic system involving the 5-quantity LSO can generate around 4,000 differentinference schemas but only 105 are valid, these being the 24 the Aristotelian ones and 81new ones. Table 1.13 shows the list of the new valid syllogisms, adhering to the capital letternotation explained above.

The validity of these new syllogisms is proved through an extension of Venn Diagrams [63]inspired by P. Geach’s work [22], which consists of an algebraic method. The first step is thedivision of Venn Diagram of a syllogism into sub-classes, as shown in Figure 1.9. Eachlower case letter (a,b, . . . ,h) denotes a subsection of the Venn Diagram and the capital letters(NT,DT and MT ) denotes the terms that configure a syllogism.

The heart of this method is to interpret each intermediate categorical statement as makinga statement about the relative size of two sub-classes. Table 1.14 shows all the algebraicinterpretations of the intermediate categorical statements for a 5-quantity LSO, where S standsfor NT and P for MT [59] and > is read as exceeds and >> as greatly exceeds. We shall


Figure I Figure II Figure III Figure IVsyllogisms containing majorities

AAT AED ATI AEDATT ADD ETO ETOATI ADO TAI TAI

EAD EAD DAOETD ETD TTIETO ETO DTO

syllogisms containing majorities and commonsAAK AEG AKI AEGATK ADG EKO EKOAKI AGO KAI KAIAKK AGG GAOEAG EAGETG ETGEKO EKOEKG EKG

syllogisms containing majorities, commons and predominantsAAP AEB PAI AEBAPP ABB EPO PAIAPT ABD BAO EPOAPK ABG GAOAPI ABO APIEAB EAB PPIEPB EPB TPIEPD EPD KPIEPG EPG PTIEPO EPO PKI

BPODPOGPOBTOBKO

Table 1.13: Moods of Intermediate Syllogistics.


ab

c

d

e

f

g

hDT

NT MT

Figure 1.9: Venn Diagram subsections.

Intermediate Proposition Algebraic Interpretationall S are P a = 0,d = 0 with b 6= 0 or e 6= 0almost all S are P b+ e >> a+d with b 6= 0 or e 6= 0most S are P b+ e > a+d with b 6= 0 or e 6= 0many S are P ¬(a+d >> b+ e) with b 6= 0 or e 6= 0some S are P b 6= 0 or e 6= 0no S are P b = 0 and e = 0few S are P a+d >> b+ e with b 6= 0 or e 6= 0many S are not-P (a+d >> b+ e) with b 6= 0 or e 6= 0some S are not-P a 6= 0 or d 6= 0

Table 1.14: Algebraic interpretation of intermediate categorical statements.

use this same version of Venn diagrams to develop our proposal of syllogistic system (seeChapter 2).

To demonstrate the validity of a syllogistic argument two alternative paths can be taken:i) deduce the algebraic representation of the conclusion from the premises; ii) show that thepremises and the denial of the conclusion are inconsistent. For instance, Table 1.15 shows anexample of intermediate syllogism proved by path ii).

The Aristotelian meta-rules (see section 1.1.1.3) are still valid for intermediate syllogisms,but they must be extended in order to involve the new schemas. The new rules are mainlyfocused on the notion the of distribution as, while in Aristotelian syllogistics there are onlytwo possible choices (distributed and undistributed), new possibilities appear in intermediate


TTI SyllogismMost DT are MT e+ f > d +gMost DT are NT d + e > g+ f

Some NT are MT b 6= 0 or e 6= 0Proof

(1)e+ f > d +g premise(2)d + e > g+ f premise

(3)b = 0 and e = 0 denial of conclusion(4) f > d +g from (1) and (3)

(5) f > d from (4)(6)d > f from (5) and (3)

(5) and (6) are contradictory

Table 1.15: TTI syllogism and proof.

syllogisms; in other words, the crisp Aristotelian concept is converted into a graduated one.Peterson proposes the concept of ‘more nearly distributed’ expressed as Distribution Index

(DI) with the following scale:

DI = 5 Subjects of universals and predicates of negatives.

DI = 4 Subjects of P and B.

DI = 3 Subjects of T and D.

DI = 2 Subjects of K and G.

DI = 1 Subject of particulars and predicates of affirmatives.

Following the numerical indices, the order from the distributed terms (DI = 5) to theundistributed terms (DI = 1) is as follows:

DI = 5 > DI = 4 > DI = 3 > DI = 2 > DI = 1

Thus, the new rules for intermediate syllogisms are the following ones:

– Distribution

R1 In a valid syllogism, the sum of DIs for the middle term must exceed 5.

R2 No term may be more closely distributed in the conclusion than it is in the premises(i.e., no term may bear a higher DI in the conclusion than it bears in the premises).


– Quality

R3 At least one premise must be affirmative.

R4 The conclusion is negative if and only if one of the premises is negative.

– Quantity

R5 At least one premise must have a quantity of majority (T or D) or higher.

R6 If any premise is non-universal, then the conclusion must have a quantity that isless than or equal to the premise.

1.1.2.3 Summary

Intermediate Syllogistics incorporates new quantifiers between the universal and particularones in syllogistic reasoning, many of them being vague. This entails a relevant innovationwith respect to the Aristotelian one; which only managed four crisp quantifiers. It also showsthat these new quantifiers also can be ordered into an LSO, although some additional con-straints must be taken into account, such as, for instance, identifying the adequate sense in theuse of the quantifiers. The authors make their proposal of syllogistic system using an LSOwith 5 quantifiers.

The incorporated innovations enable Peterson and Thompson to generate a vast number ofnew syllogisms that contributes to expand the expressive power of the Aristotelian Syllogis-tics. However, it still limited to proportional quantifiers (others, such as exception or absolutequantifiers, are not considered) and arguments constituted by three terms, two premises anda conclusion. Many arguments involving quantities that human beings use in their daily livesare far beyond the scope of this framework.

It is worth noting that the algebraic method based on Venn Diagrams will be used as thebase for our proposal of syllogistic system compatible with TGQ.

1.1.3 Exception Syllogistics

As we have already stated, in natural language there are other kinds of quantifiers different toproportional ones, such as exception (“all but eight”, etc.) or comparative (“double”, “threemore”, etc.), which human beings also use in their inferences, even though they are not con-sidered in syllogistics. We have shown that Intermediate Syllogistics significantly increases


Traditional formulation Exception formulation SymbolizationAll S are P All but 0 S are P 0SAPNo S are P At most 0 S are P 0SEPSome S are P At least 1 S is P 1SIPSome S are not P At least 1 S is not P 1SOP

Table 1.16: Recasting A, E, I and O as exception statements.

the number of quantifiers of Aristotelian Syllogsitics, but considering exclusively proportionalones, such as “most”, “many”, “few”, etc.

In [39], W. Murphree proposes a new approach to syllogistic reasoning based exclusivelyon exception quantifiers (i.e.; “all but three”, “all but five”, etc.) and on categorical statementswith the basic form all but x S are P, where x stands for a N. In this way, the proportionalreading of the quantifiers is substituted by an absolute one, as it involves a direct referenceto an absolute value. Consequently, new schemas of syllogism emerge, one of the principlenovelties of which is the consideration of several possible conclusions instead of one singleone.

This section is organized as follows: section 1.1.3.1 addresses the theory of exceptioncategorical statements, describing how quantified statements are interpreted in terms of ex-ception quantifiers; section 1.1.3.2 describes the characteristics of the LSO; section 1.1.3.3,describes the reasoning patterns and the procedure for inferring the conclusion and, finally,section 1.1.3.4 concludes with a summary analysing the strengths and weaknesses of thisframework.

1.1.3.1 Theory of Exception Categorical Statements

W. Murphree assumes that a categorical statement can be expressed using exception quanti-fiers, whose basic form is all but x, with usually x ∈N. For instance, “all human beings aremortal” is equivalent to “all but 0 human beings are mortal”, where the numerical value 0indicates the “exception” of the quantifier. However, depending on the quantifier under con-sideration and its associated presuppositions, the basic form of exception quantifiers can bemodified. Table 1.16 shows the categorical exception statements equivalent to the four tradi-tional categorical statements (i.e.; A, E, I and O). For instance, “some” is interpreted as “atleast one” as it entails the existential import.


Exception formulation SymbolizationAt least all but x S are P xSAPAt most x S are P xSEPAt least x S is P xSIPAt leas x S is not P xSOP

Table 1.17: General form of exception statements.

Intermediate quantifiers also can be generated using this form, by modifying the valueof x, which goes from 0 to |S|. Table 1.17 shows the general form exception statements forintermediate quantifiers depending on the universal and particular ones. It is worth noting thatthis formulation can require the assignation of a cardinality to S as, otherwise, the transforma-tion cannot be properly executed. For instance, let us consider “few students are tall”. If weassume |S|= 10 and that “few” means “no more than three”, we can use xSEP symbolizationand say “at most three students are tall”.

On the other hand, the introduction of the numerical value x allows Murphree to talk aboutstatements in terms of strength and weakness. Thus, statements with “the same quantity andquality are stronger or weaker than another according to their exception values” [39, p.51].For universal statements (xSAP and xSEP), if the exception value increases, then the claim isweaker; in the case of particular ones (sSIP and xSOP), if the exception value increases, thenthe claim is stronger. For instance, “at most ten students are tall” is stronger than “at mostfive students are tall” or “at least five dogs are barking” is weaker than “at least ten dogs arebarking”.

Regarding the distribution property, this does not have modifications with respect to Inter-mediate Syllogistics (see Table 1.2).

1.1.3.2 Logic Square of Opposition

The introduction of new numerical values in the quantifiers entails a new way of building theLSO. All its relationships are preserved, although Murphree focuses on the so-called equiv-alent relationships: conversion, obversion and contraposition. Table 1.18 shows the relation-ships among these properties. Subalternation is explained in terms of strength and weakness;the strongest statement entails the weakest one.

Nevertheless, the problem of existential import is not avoided with categorical statementsof this type. Although Murphree [39, p.52] explicitly considers this question, he rejects it


Original Conversion Obversion ContrapositionxSAP xPAS xSEP xPASxSEP xPES xSAP xPESxSIP xPIS xSOP xPISxSOP xPOS xSIP xPOS

Table 1.18: Equivalent relationships between exception statements.

because, in general, the subject-term of universal statements is not necessarily not empty. Thegeneralization of this idea is defined by two observations:

1. A statement is inherently existential to the magnitude of x if and only if the existenceof at least x S is a precondition for the truth of the statements.

2. The existence of at least x S is a precondition for the truth of xSIP and xSOP, but xSAPand xSEP are compatible with the existence of any number of S whatsoever.

For instance, a statement with the form at least x S are P entails that |S| > 0 to be true;while, at most x S are P only entails |S| < x which is compatible with |S| = 0. Neverthe-less, the meaning of exception value x demands the existence of a number of elements in S

and, hence, a number of presuppositions appear. Murprhee distinguishes between minimum

presuppositions and maximum presuppositions.Minimum presuppositions belong to particular statements and are expressed as at least x.

For instance, if we consider |S| ≤ 10, “all but 0 S is P” (0SAP) entails “at least 10 S are P”(10SIP); and in addition;

1SAP entails 9SIP by presupposition,2SAP entails 8SIP by presupposition,

. . .9SAP entails 1SIP by presupposition,

On the other hand, 9SIP also implies 8SIP, 7SIP, etc. and, therefore, they are also en-tailed by 1SAP. Thus, assuming that S ≥ y, xSAP implies y− xSIP where y and x are thevalue of exception in particular and universal statement, respectively. In this way, minimumpresuppositions give existential import to universal statements.

Maximum presuppositions are expressed as at most x. Thus, let us now consider |S| ≥ 10,“at least 1 S is P” (1SIP) entails “all but 9 S are P” (10SAP); and in addition;


1SIP entails 9SAP by presupposition,2SIP entails 8SAP by presupposition,

. . .10SIP entails 0SAP by presupposition,

As in the minimum presuppositions, 10SAP also implies the weaker claims 9SAP, 8SAP,etc. and, therefore, they are also entailed by 10SIP. To avoid any possible incongruousness,the values of the exception in particular statements (denoted by y) must be complementary tothe value of exception in universal statements (denoted by x) regarding S; i.e., S≤ y+ x.

Four different considerations can be made regarding minimum and maximum presuppo-sitions [39, p. 55-56]:

1. Universal statements entail particular statements by virtue of minimum presuppositionsand particular ones entail universal ones by maximum presuppositions.

2. Minimum presuppositions concern the distributed terms in universal statements andmaximum presuppositions concern the undistributed ones in particular propositions.

3. Minimum presuppositions are not a truth criterion either for particular or for universalstatements, while maximum presuppositions can render universal ones true and partic-ular ones false.

4. The cardinality of S is a relevant datum. Assuming some undetermined minimum car-dinality of S is insufficient for making inferences; however, every universal propositionfollows from some maximum cardinality of S; for instance, from at most x S, we candeduce xSEP and xSAP for any P.

The combination of minimum and maximum presuppositions is a necessary condition tobuild the LSO based on exception statements [39, p. 56-57]. Thus, from xSAP, xSEP, xSIP,xSOP (see Table 1.17), minimum presuppositions allow us to infer, for instance, xSIP andxSIP from xSAP; maximum presuppositions supports the inference of xSAP from secondaryimplications of the particular ones. In addition to presuppositions, the derivations also requirethe properties of obversion (Ob), conversion (Cv) and contraposition (Cp) (see Table 1.18).Table 1.19 shows the derivation of aSIP.


1.aSIPPresuppositions 2.S≤ a+ x

3.S≥ b+ y4.P≤ a+ y5.P≥ c+ x

Derivations 6.xSAP Ps 1,27.aPIS Cv 18.yPAS Ps 4,79.ySAP Cp 810.bSIP Ps 3,911.xPAS Cp 612.cPIS Ps 5,1112.cSIS Cv 12

Table 1.19: Equivalent relationships between exception quantifiers.

1.1.3.3 Theory of Exception Syllogism

Arguments built with exception statements have the same form as the previous frameworks;i.e., syllogisms constituted by three terms, two premises and a conclusion. However, thissystem admits multiple alternative valid conclusions. It is an immediate consequence of inter-preting quantifiers using absolute numerical values in combination with the presuppositionsexplained above.

All Aristotelian moods have been checked in this framework and they are valid, attendingto the equivalence established between traditional categorical statements and exception ones(see Table 1.16).

Regarding the procedures for checking the validity of the syllogisms, only syllogisticrules [39] and the algebraic method based on the use of Venn diagrams [38] are considered.Hence, the new rules to be added depend on two possible situations: i) inferences withoutusing presuppositions and ii) inferences based on presuppositions.

In case i), two new rules appear [39, p.58]:

– R1 The exception for a universal conclusion must not be less than the sum of the ex-ceptions in the premises.

– R2 The exception for a particular conclusion must not be greater than the exception ofthe particular permise minus the exception of the universal premise.


MP xMTANTNP aNT IMTC a− xNT IMT

Table 1.20: Datisi syllogism with exception propositions.

R1 generates three possible scenarios:

– Invalid argument: the value of the exception in the conclusion is less than the additionof the value of the exceptions in the premises.

– Valid argument: the exception of the conclusion is greater than the exception of thepremises.

– The strongest valid argument: the value of the exception in the conclusion equals theexception of the premises.

R2 generates these same three possible scenarios but for arguments with a particular con-clusion. For instance, let us consider AII Mood from Figure I (see Table 1.20). The quantifierof the conclusion (C) is a−x, where x denotes the value of the universal premise and a standsfor the value of the particular one. The problem of the existential import is avoided substitut-ing the particular premise (xNT IDT ) by a+ xNT IDT and the conclusion by aNT IMT .

Exception Syllogistics define 64 syllogistic schemas, 40 of which are new (the remaining24 are the Aristotelian ones). Table 1.21 shows all of them. The conclusion does not appearbecause there are multiple valid alternatives, which, as we have already stated, depend on thepresuppositions assumed and the cardinality of the involved terms.

Furthermore of the division into Figures, these patterns are classified into four groupsaccording to the following rules: i) at least one premise is universal; ii) the occurrences of DTin the premises have opposite distribution values. Syllogisms of Group 1 are the only onesfully valid as they satisfy both rules and four conclusions can be inferred: the Aristotelian oneand its three corresponding equivalent statements. Group 2 comprises syllogisms that onlysatisfy condition i) and, hence, require an additional presupposition (minimum of maximum)for the DT to satisfy rule ii). Group 3 are syllogisms that only satisfy rule ii), the DT isproperly distributed but the premises are not right; in this case, a maximum presuppositionfor the extreme terms (NT or MT) compatible with the distribution of DT must be assumed tosatisfy rule i). Finally, Group 4 comprises syllogisms that do not satisfy any of the two rules;


Group Fig. I Fig. II Fig. III Fig. IV Satisfy1 AA AE AI AA i), ii)

AI AO AO AEEA EA EI EIEI EI EO EOIE IE IA IAIE IE IA IAOE OA IE IE

OAOE

2 AE AA AI i)AO AI AOEE EE AA EAEO EO AE EEIA IA EE OAOA OE IE OE

3 IO IO OI ii)OO OI OO

4 II noneIO

II II OI IIOI OO OO IO16 16 16 16

Table 1.21: Moods of Exception Syllogistics.

to satisfy them, it is necessary to assume a maximum presupposition for the DT , which mustbe an universal proposition where DT is the subject, being thus distributed.

Venn diagrams [64, 38] are also valid for representing arguments of Exception Syllogis-tics and it is not necessary to change the notation respect to Intermediate Syllogistics (seesection 1.1.2).

In [64], N. Pfeifer proposes an initial attempt to combine exception and intermediate quan-tifiers into a single statement with the following form intermediate statement and exception

statement. Statements of this type are also valid for building syllogisms. Table 1.22 shows anexample of the intermediate syllogism TTI from Figure III [64, p.69]5.

5In this example, the verb to be is substituted by to have for simplicity, since “most birds have wings” is equiva-lent to “most birds are issuing of wings”.


MP Most birds have wings and at least all but x birds have wingsNP Most birds have feathers and at least all but y birds have feathersC More than x and more than y things have feathers and wings

Table 1.22: TTI syllogism combining intermediate and exception quantifiers.

The procedure for checking the validity of these syllogisms is the algebraic method incombination with Venn diagrams. The first step is the formalization of the premises accordingto Figure 1.9, obtaining:

MP≡ e+ f > x≥ d +g

NP≡ d + e > y≥ f +g(1.3)

Applying the algebraic method explained in section 1.1.2, we obtain the indirect proof:

1 ¬(e > x) (indirect proof assumption)

2 ¬(e+ f > x) (1)

3 e+ f > x (MP)

4 e > x (1)

5 ¬(e > y) (indirect proof assumption)

6 ¬(d + e > d + y) (5)

7 ¬(d + e > y) (6)

8 d + e > y (NT)

9 e > y (indirect proof (5)-(8))

10 e > xande > y (4 and 9)

11 e > max(x,y) (10)

12 b+ e > max(x,y) (conclusion)

(1.4)

N. Pfeifer show us that the combination of different type of propositions in a single state-ment also can generate valid syllogisms, which, furthermore, are more informative than thetypical ones.

1.1.3.4 Summary

Exception categorical statements introduce a new way of defining the space between the uni-versal and the particular propositions. In this case, the quantifiers refer to a number of ele-


ments of the universe of reference that do not fulfil the property of the predicate-term (theso-called exception quantifiers) and they only have an absolute reading, an option not consid-ered in the models previously described.

From the point of view of reasoning, this is the first framework that deals with absolutequantities in syllogisms. This opens a new conception regarding inference schemas, where thereasoning process is heuristic (i.e., selecting a conclusion among different valid conclusions),rather than probative (i.e., checking if a concrete conclusion follows from the premises). Nev-ertheless, on the other hand, additional assumptions not explicitly considered in the argumentmust be assumed for reasoning, such as the described maximum and minimum presupposi-tions or knowing the size of S.

In conclusion, Exception Syllogistics gives rise to relevant challenges in syllogistic rea-soning, such as the use of non-proportional quantifiers or heuristic reasoning, although theproblem of vagueness is sparingly analysed.

1.1.4 Interval Syllogistics

All the previous approaches extend Aristotelian Syllogistics introducing new quantifiers tomake a subdivision of the space between the universal and the particular statements, eitheras proportional quantifiers (such as intermediate ones) or absolute ones (such as exceptionones). Most of them are vague and, hence, have a context-dependent meaning. Nevertheless,both Intermediate Syllogistics and Exception Syllogistics consider only crisp definitions forquantifiers.

The main approaches for dealing with vague intermediate quantifiers have been proposedin the field of fuzzy logic. The concepts of fuzzy set and fuzzy number open up a newframework for managing syllogistic reasoning involving vague intermediate quantifiers usingfuzzy definitions for them. In [17], D. Dubois and H. Prade introduce a distinction betweenimprecise quantifiers (such as “between 30% and 40%”, etc.), which are defined as intervals,and fuzzy ones (such as “most”, “few”, etc.), which are defined using fuzzy numbers. Thus,preserving the typical form of categorical statements Q S are P, both type of vague quantifiers(which are proportional) are substituted by intervals expressed as percentages, or by fuzzynumbers defined through trapezoidal functions. Nonetheless, in subsequently papers [18, 3,16], they only focus on syllogistic patterns involving interval quantifiers, not fuzzy ones, and,for that reason, we shall deal with them first.


This framework also incorporates three significant changes with respect to the previousapproaches to syllogism: i) the concept of intermediate quantifier is substituted by a numericalpercentage; ii) as a consequence, the eminent linguistic character of syllogism is substitutedby a numerical one; and, iii) the reasoning process, based mostly on logical proofs, becomesa calculation procedure, endowing syllogism with a heuristic character. On the other hand,it is relevant to note that the classical models of syllogisms are not explicitly dealt with and,hence, their compatibility is not checked.

This section is organized as follows: section 1.1.4.1 addresses the theory of interval cate-gorical statements, describing how quantified statements are interpreted in terms of intervals;section 1.1.4.2, describes the path for allocating an interval to a linguistic label and how theyare structured; section 1.1.4.3, describes the reasoning patterns and the procedure for calcu-lating the quantifier of the conclusion and, finally, section 1.1.4.4 concludes with a summaryanalysing the strengths and weaknesses of this proposal.

1.1.4.1 Theory of Interval Categorical Statements

We initially introduce the concept of interval categorical statement to refer to the statementsmanaged by this approach. Typical examples are “between 30% and 50% students are blond”or “a half of the students are blond”. As we can see, they preserve the typical form of categor-ical statements Q S are P, where Q denotes the interval quantifier “between 30% and 50%” orthe linguistic label “a half of”, S stands for the term-set students and P by the term-set blond.It is worth noting that only proportional quantifiers and affirmative terms are dealt with in thisframework; the treatment of the negation of quantifiers or terms is not addressed.

Regarding the precision of the quantifiers, Dubois et al. distinguish three definitions [17]:

1. Imprecise: they denote proportional quantities that are not precisely known but haveprecise bounds. They are usually asserted by expressions taking the form between q

and q, less than q, more than q,. . . where q,q ∈ R. For instance, “between 30% and50%” (see Figure 1.10) or “less than 70%”. They are represented as Q =

[q,q].

2. Precise: they denote proportional quantities that are precisely known and also haveprecise bounds. They are usually expressed by exact percentages such as “30%” (seeFigure 1.11), “70%”, etc. In this case, the lower and upper bounds of the intervalcoincide (q = q).


0

0.2

0.4

0.6

0.8

1

10% 20% 30% 40% 50% 60% 70%

Percentage

Figure 1.10: Graphic representation of “between 30% and 50%”.

0

0.2

0.4

0.6

0.8

1

10% 20% 30% 40% 50% 60% 70%

Percentage

Figure 1.11: Graphic representation of “30%”.

3. Fuzzy: they denote proportional quantities that are vaguely known and whose boundsare also fuzzy. These are the properly called fuzzy quantifiers. They are usually ex-pressed by percentages modified by expressions such as “around”, “more or less”, etc.or by linguistic terms such us “most”, “many”, “few”, etc. For instance, “around40%” (see Figure 1.12), “aproximately 70%”, etc. They are represented by fuzzynumbers approximated through the usual four points trapezoidal representation Q ={

q∗,q,q,q∗}

, where KERQ = [q,q] corresponds with the kernel and SUPQ = [q∗,q∗]

with the support. Figure 1.12 shows the graphical form for the quantifier “around40%”={0.25,0.33,0.47,0.55} with SUParound40% = [q∗ = 0.25,q∗ = 0.55] andKERaround40% = [q = 0.33,q = 0.47].


0

0.2

0.4

0.6

0.8

1

10% 20% 30% 40% 50% 60% 70%

Percentage

Figure 1.12: Graphic representation of “around 40%”.

1.1.4.2 Lattices of Linguistic Labels

As explained in the previous sections, the LSO is built over the relationships of oppositionand it is fundamental to prove the validity of syllogisms. However, in Interval Syllogistics thequantifiers are numerically interpreted (in terms of intervals or in terms of fuzzy numbers) butwithout considering opposition relationships. Therefore, the development of an LSO here isnot possible.

Nevertheless, in [16], authors propose a qualitative version of syllogism managing a setof elementary labels (there are seven: “none”, “almost none”, “few”, “about half”, “most”,“almost all”, “all”) of linguistic quantifiers where each label corresponds to an interval [a,b]⊆[0,1]. These labels are organized according to the concept of order, attending to two dimen-sions: specificity and certainty. This allows authors to build a biordered structure in tree form,which is reproduced in Figure 1.13. The allocation to each quantifier of its correspondingsubinterval follows the path described below.

Let P be a partition of [0,1] in adjacent subintervals and let P also be, by convention,the linguistic scale and the partition of the quantifiers (the seven ones mentioned above). Itis also assumed that the linguistic scale is symmetric respect to 0.5 (which is reasonable)and the concept of antonymy is introduced. Although antonymy is an opposition relationshipequivalent (in some sense) to the relationships of contrary and subcontrary in the classicalLSO (see Figure 1.1) or to inner negation in the Modern LSO (see Figure 1.2), it is notintroduced as a criterion for the initial selection of the quantifiers but as a condition for theselection of the numerical intervals, since it is defined as ANT ([q,q]) = [1− q,1− q]. Thus,every quantifier has an antonym that also belongs to the scale and the partition; hence, it is


Almostall

None Almostnone

Few Abouthalf

Most All

[None Al-none] [Al-none Few] [Few Ab-half] [Ab-half Most] [Most Al-all] [Al-all All]

[None Few] [Al-none Ab-half] [Few Most] [Ab-half Al-all] [Most All]

[None Ab-half] [Al-none Most] [Few Al-all] [Ab-half All]

[None Most] [Al-none Al-all] [Few All]

[None Al-all] [Al-none All]

[None All]

Specificityordering

Certainty ordering

Figure 1.13: The structure ordered of linguistic quantifiers.

a fully symmetric relationship. For instance, ANT (none) = all or ANT (most) = f ew (seeFigure 1.14).

To determine the corresponding subintervals to the pointed out seven linguistic quantifiers,we must take two parameters a,b ∈ [0,1] where a < 0.5,b < 0.5 and a < b; thus, a = q andb = q. The [0,1] interval can be (non-strictly) symmetrically partitioned, as shown in 1.5,and each interval is assigned to a linguistic label. Table 1.23 shows this allocation and thecorresponding result for a = 0.2 and b = 0.4.

P = 0,(0,a], [a,b], [b,1−b], [1−b,1−a], [1−a,1),1 (1.5)

The interpretation of relationships between the linguistic quantifiers in terms of orderinstead of terms of opposition or negation is a new perspective in the analysis of categoricalstatements and, hence, also of syllogistic reasoning. However, it is also true that the key pointof Interval Syllogistics is numerical quantifiers instead of linguistic ones, whose interpretationin terms of negation is closer to their use in natural language than those based on the order.

Another question that must be considered in the allocation of numerical values to linguis-tic terms is the so-called problem of embarrassment of richness in modelling vague naturallanguage quantifiers [25]. The main idea is that there is a huge space of candidates for truth


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Percentage

Few Most

Figure 1.14: Antonymy relationship between “few” and “most” in a interval scale.

Linguistic quantifier Interval Numerical intervalNone 0 0Almost none (0,a] (0,0.2]Few [a,b] [0.2,0.4]About half [b,1−b] [0.4,0.6]Most [1−b,1−a] [0.6,0.8]Almost all [1−a,1) [0.8,1)All 1 1

Table 1.23: Correspondence between the linguistic terms, their intervals in the scale and thevalues when a = 0.2 and b = 0.4.

functions that may be acceptable for defining a linguistic quantifier and it is excessively com-plex to manage. For instance, let us consider again the example “a half of the students areblond”. Which is the interval that best defines the quantifier? According to Table 1.23 it is a

half of= [0.4,0.6]. However, if the elements in set of basic linguistic quantifiers are increased(for instance; from seven labels to nine) or the values of a and b changes or the intervals havea certain degree of overlap, then the interval allocated to “about half” also changes. Thus,the critical question is determining what the most accurate definition is or the location of theborder over which the values are not acceptable. Although it is a significant problem, it isoutside the framework of [16] and, for that reason, it is beyond the scope of our analysis.


1.1.4.3 Theory of Interval Syllogism

The previously analysed theories of inference focused mainly on identifying the valid infer-ences schemas through logical proofs, although Exception Syllogistics pointed out an heuris-tic aim for the arguments. However, the perspective of analysis changes in this framework.As shown in the previous section (1.1.4.2), the LSO is substituted by a tree structure basedon the concept of order over a set of more or less arbitrarily chosen linguistic labels; and thelogical proofs for a calculation procedure which aims to determine the numerical value of thequantifier of the conclusion in terms of the quantifiers of the premises. Hence, the premisesconstitute a set of restrictions over which the bounds of the quantifier of the conclusion mustbe calculated, maximizing the upper bound and minimizing the lower one; in other words, thereasoning process consist of obtaining the most favourable and most unfavourable proportionsamong the terms of the conclusion according to the proportions expressed in the premises.

Depending on the positions of the terms in the premises and the form of the conclusion,three reasoning patterns are defined [17, 18]: Pattern I, Pattern II and Pattern III. Althoughall of them involve three terms, as the classical models, the roles of MT , NT and DT are notconsidered. Only Pattern I preserves this structure, while Pattern II and Pattern III [18] havea different mechanism involving the logic operation of conjunction. In [79], both kinds ofinference mechanisms are described:

1. Property inheritance (asymmetric syllogism): A term-set X and a term-set Z are linkedvia the concatenation of X with a term-set Y and Y with Z:

PR1 Y in relation to Z

PR2 X in relation to Y

C X in relation to Z

This schema matches up with Figure I of Aristotelian Syllogistics. The other threefigures change the position of Y in the premises.

2. Combination of evidence (symmetric syllogism): The ties between X and Z and be-tween Y and Z are calculated separately and both are joined in the conclusion by a logicoperator (conjunction or disjunction):


(a) Schema

Q1 As are Bs;Q′1 Bs are AsQ2 Bs are Cs;Q′2Cs are BsQ As are Cs;Q′ Cs are As

(b) Example

[0.85,0.95] students are young[0.25,0.35] young people are students[0.90,1] young people are single[0.60,0.80] single people are young[0.51,1] students are single[0.1,0.36] single people are students

Table 1.24: Pattern I.

PR1 Y in relation to Z

PR2 X in relation to Z

C X&Y in relation to Z

This schema is not considered in the classical approaches to syllogism as it is not basedon the use of a middle term.

So, let A,B and C be the term-sets involved in a syllogism and Q1, Q′1, Q2, Q′2, Q and Q′

the possible quantifiers (which can be precise, imprecise or fuzzy).Pattern I is constituted by three pairs of premises and one pair of conclusions (see Ta-

ble 1.24(a)) where Q1 denotes the quantifier of the first premise and Q′1 its converse, Q2 de-notes the quantifier of the second premise and Q′2 its converse, and Q and Q′ denote quantifiersfor the conclusions. This is illustrated by the example in Table 1.24(b)6.

Figure 1.15 shows a graphic representation of the Pattern I schema. The representationusing Venn diagrams is substituted by a graph in which the circles denote the term-sets A,B

and C; arcs with solid lines denote the quantifier Qi for that premise; arrows with dotted linesdenote the quantifiers for the conclusions; that is, Q and Q′.

Pattern I belongs to asymmetric syllogisms since B plays the role of the middle term; i.e.,the link between the subject (A) and the predicate (C) of the conclusion is via concatenationof A with B and B with C. It is also worth noting that the links between all the three involvedpredicates have to be fully known.

6The data for this argument were taken from [16].


A B

C

Q'1

Q2

Q1

Q'2

Q

Q'

Figure 1.15: Graphic representation of Pattern I.

For precise quantifiers, the quantifier of the conclusion Q :=[q,q]

is calculated using thefollowing expressions:7

q = q1·max(

0,1− 1−q2

q′1

)(1.6)

q = min(

1,1−q1 +q1·q2

q′1,

q1·q2

q′1·q′2,

q1·q2

q′1·q′2[1−q′2 +q1]

)(1.7)

For imprecise quantifiers, expression (1.6) should be minimized and (1.7) maximized,obtaining:

q = q1·max

(0,1−

1−q2

q′1

)(1.8)

q = min

(1,1−q

1+

q1·q2

q′1

,q1·q2

q′1·q′

2

,q1·q2

q′1·q′

2

[1−q′2+q1]

)(1.9)

As has already been pointed out, the management of fuzzy quantifiers only appears in [17].It consists of independently calculating the intervals corresponding to the support ([q∗,q∗])and to the kernel ([q,q]) and, subsequently, building the corresponding fuzzy number. Never-theless, as pointed out in [56], this approach generates problems when non-normalized resultsare obtained as the truth function of the kernel is less than 1.

7Calculation of Q′ is identical. For details of the proof, the reader is referred to [17, 18].


(a) Schema

Q1 As are BsQ′1 Bs are AsQ2 Bs are CsQ3 As are CsQ As and Bs are Cs

(b) Example

[0.85,0.95] students are young[0.25,0.35] young people are students[0.90,1] young people are single[0.2,1] students are single[0.9,1] young students are single

Table 1.25: Pattern II.

Figure 1.16: Graphic representation of Pattern II.

Pattern II is constituted by three terms and three premises (two of them being converse)and a single conclusion, which includes the three terms. Table 1.25(a) shows its schema forthe quantifiers Q1, Q′1, Q2 and Q3 and Q is calculated in terms of them. Table 1.25(b) showsan illustrative example.

For precise quantifiers, the expressions that define the bounds of Q are the following ones:

q = max(

0,1− 1−q3

q1,1− 1−q2

q′1

)(1.10)

q = min(

1,q3

q1,

q2

q′1

)(1.11)


Q1 As are Bs Q′1 Bs are AsQ2 Bs are Cs Q′2 Cs are BsQ3 As are Cs Q′3 Cs are As

Q As and Bs are Cs

Table 1.26: Pattern II’.

Figure 1.17: Graphic representation of Pattern II’.

For imprecise quantifiers, the previous expressions must be maximized and minimizedrespectively, obtaining:

q = max

(0,1−

1−q3

q1

,1−1−q

2q′

1

)(1.12)

q = min

(1,

q3

q1

,q2

q′1

)(1.13)

If additional information is available, Pattern II has a general version called Pattern II’.It is obtained by adding the premises Q′2 Cs are Bs and Q′3 Cs are As to the schema of Ta-ble 1.25(a), obtaining the schema shown in Table 1.26 and the graph of Figure 1.17.

For precise quantifiers, the bounds for Q are:

q = max(

0,1− 1−q3

q1,1− 1−q2

q′1,

q3

q1+

q2

q′1·1− 1

q′2

)(1.14)

q = min(

1,q3

q1,

q2

q′1

)(1.15)


(a) Schema

Q′2 Cs are BsQ′3 Cs are AsQ Cs are As and Bs

(b) Example

[0.3,0.5] single people are young[0.7,0.9] single people are students[0,0.5] single people are young and students

Table 1.27: Pattern III.

A B

C

Q

Q'3 Q'2

Figure 1.18: Graphic representation of Pattern III.

For imprecise quantifiers, the formula of the lower bound must be minimized and theformula of the upper bound must be maximized:

q = max

(0,1−

1−q3

q1

,1−1−q

2q′

1

,q

3q1

+q

2

q′1·1− 1

q′2

)(1.16)

q = min

(1,

q3

q1

,q2

q′1

)(1.17)

Pattern III is constituted by only two premises and a conclusion, which also includes thethree terms. Table 1.27(a) shows its schema and Table 1.27(b) an example.

For precise quantifiers, the bounds for Q are:

q = max(0,q′3+q′2−1

)(1.18)

q = min(q′3,q′2

)(1.19)


Q1 As are Bs Q′1 Bs are AsQ2 Bs are Cs Q′2 Cs are BsQ3 As are Cs Q′3 Cs are As

Q Cs are As and Bs

Table 1.28: Pattern III’.

A B

C

Q

Figure 1.19: Graphic representation of Pattern III’.

In the case of imprecise quantifiers, the extension is obvious, as in the previous case:

q = max(

0,q′3+q′

2−1)

(1.20)

q = min(q′3,q

′2)

(1.21)

As in the case of Pattern II, there is a general version of this schema called Pattern III’.We can say that Pattern III’ is complementary to Pattern II’ as they share the same set ofpremises and the conclusion of the first one is the converse of the conclusion of the second one.The complete schema is shown in Table 1.28. Figure 1.19 shows the graphical representationthereof, where Q is calculated in terms of Q1,Q′1,Q2,Q′2,Q3 and Q′3.

For precise quantifiers, the lower and the upper bounds for Q are the following:

q = max(

0,q′3 +q′2−1,q′3q3· (q1−1)+q′3,

q′2q2· (q′1−1)+q′2

)(1.22)

q = min(

q′3,q′2,q1 ·

q′3q3

)(1.23)


For imprecise quantifiers, the extension of the lower bound is less simple than in theprevious cases, for two reasons: a) the value for q

3+ q

1+ 1 can be positive or negative and

according to it, we must use q′3

or q′3 to compute the lower bound and obtaining positivevalues; the same is true for q′1,q2 and q′2; b) to calculate the lower bound in this pattern onlyfive quantifiers are needed, not six; therefore, these six expressions must be used to tightenthe bounds. Nevertheless, the extension for the upper bound is:

q = min

(q′3,q

′2,q1 ·

q3

q′3

,q′1 ·q2

q′2

)(1.24)

1.1.4.4 Summary

One of the main contributions of this framework is to deal with the intrinsic vagueness ofintermediate quantifiers directly instead of transforming them into precise definitions, as hap-pens, for instance, in Exception Syllogistics (see section 1.1.3). Regarding the theory ofinference, it is relevant to note that the reasoning process is a heuristic method for calculatingthe quantifier of the conclusions according to the restrictions of the premises rather than to theelaboration of a logical proof.

The authors also propose the use of subintervals of [0,1] and the allocation thereof to aquantifier partition whose members are organized over the concepts of specificity and cer-tainty, thus generating a biordered structure. The role assigned to this lattice is comparablewith the role of the LSO in the other syllogistic proposals, insofar as between the number oflinguistic quantifiers that can be managed is limited and they have some kind of links betweenthem. Their differences reside in the type of relationships considered: the LSO is based onrelationships of opposition, while the lattices of linguistic labels are based on relationships ofspecificity, certainty and antonymy.

Our contribution to this framework is to check its compatibility with Aristotelian Syllo-gistics [51]. Only Pattern I adheres to the structure of asymmetric syllogisms and only FigureI is compatible for the conclusion Q, as Table 1.29 shows, owing to the position of the middleterm. Figures II, III and IV cannot be managed as all patterns have the middle term in thesame position. Thus, if we analyse the six moods that constitute Figure I using consistent def-initions for the four classical quantifiers (A(all) := [1,1], E (none) := [0,0], I (some) := [ε,1],O(not all) := [0,1− ε]), with ε ∈ (0,1], all of them exhibit correct behaviour (see Table 1.30).

A further weak point of Interval Syllogistics is that it only manages proportional quan-tifiers. Other quantifiers, such as exception or comparative ones, are avoided. On the other


Pattern Major premise Minor premiseFigure I Subject PredicateFigure II Predicate PredicateFigure III Subject SubjectFigure IV Predicate SubjectPattern I Subject Predicate

Table 1.29: Position of the middle term in the Aristotelian figures and Pattern I.

AAA EAE AII EIO AAI EAOPattern I Yes Yes Yes Yes Yes Yes

Table 1.30: Behaviour of Pattern I with respect to the six moods in Figure I.

hand, the use of fuzzy numbers or fuzzy intervals to represent the meaning of the quantifiershas not been developed in the literature. Despite this, Interval Syllogistics constitutes a solidsupport for the development of framework for syllogistic reasoning compatible with the TGQ.

1.1.5 Fuzzy Syllogistics

L. Zadeh [90, 91, 93] and R. Yager [88] developed a genuine model of fuzzy syllogism, wherequantifiers are always interpreted as fuzzy numbers (no intervals) and manipulated using fuzzyarithmetic, which represents a further step with respect to the previous approaches. Theyassume that the concept of quantifier is directly linked with the concept of cardinality and,more generally, with the concept of measuring the cardinality of a fuzzy set [93], expressedby means of Σ− count cardinality [94]. Hence, the management of quantified statementsinvolves the use of a fuzzy cardinality measure and fuzzy numbers.

The inference schemas preserve the basic structure of the Aristotelian ones; i.e, threeterms, two premises and a conclusion. However, as in Interval Syllogistics (see section 1.1.4),there are models based on the chaining by a middle term (asymmetric syllogisms) and modelsbased on the combination of evidence (symmetric syllogism)8.

Given that fuzzy quantifiers must be manipulated through fuzzy arithmetic [93], the log-ical analysis of the arguments is substituted by an arithmetic one. This idea is in the samevein as the algebraic approaches to syllogistic reasoning, but using the fuzzy version thereof.

8They are explained in section 1.1.4.3


This implies a reasoning process, of a heuristic type, where the quantifier of the conclusion iscalculated according to the fuzzy numbers of the premises.

On the other hand, it is worth noting that classical syllogistic proposals are not explicitlyconsidered as a particular case, and the compatibility between them has not previously beenchecked. Our contribution to this framework is to analyse this question [50, 51].

This section is organized as follows: section 1.1.5.1 describes the form of quantified state-ments and how the linguistic quantifiers are interpreted as fuzzy numbers; section 1.1.5.2shows the fuzzy syllogistic schemas and, finally, section 1.1.5.3, summarize the strengths andthe weakness of this framework.

1.1.5.1 Theory of Fuzzy Categorical Statements

The standard form of fuzzy statements for syllogistic reasoning is the typical one; i.e, Q S are

P, being Q a fuzzy quantifier, S the subject-term and P the predicate-term. For this reason, wecall these statements fuzzy categorical statements.

Zadeh distinguishes two kinds of linguistic quantifiers: first type or absolute quantifiers,and second type or proportional quantifiers. First type refers to absolute quantities, thosewhose value is independent of the number of elements in the subject-term S; for instance,“around five”, “many”, “some”, etc. Second type refers to relative quantities, those whosevalue is relative to S; for instance, “many”, “more than a half”, “most”, etc. Thus, “around

ten students are blond” is an absolute fuzzy categorical statement and “most of students areblond” is a proportional fuzzy categorical statement. It is worth noting that some linguisticquantifiers (e.g. “many”) can be absolute or proportional and, therefore, belong to one typeor other according to the context [44].

The main point of Zadeh’s quantification framework is the identification between linguis-tic quantifiers and fuzzy numbers [90]. Hence, absolute quantifiers are interpreted as absolutefuzzy numbers and proportional quantifiers are interpreted as proportional fuzzy numbers.This is linked with the cardinality of the term-sets (fuzzy or not) over which it is quantified.In the case of fuzzy sets, Zadeh proposes a fuzzy cardinality measure, Σ−Count [94]9, toestablish that link.

9Σ−Count generalizes the classical cardinality c of a subset A ⊂ E; that is c(A) = Σe∈E IA(e) where IA is themembership of e to A and

IA(e) ={

1 if e ∈ A0 if e /∈ A


(P1) Most students are young(P2) Most young students are single(C) Most2 students are young and single

Table 1.31: An example of Zadeh’s fuzzy syllogism.

Q1 As are BsQ2 Cs are DsQ Es are Fs

Table 1.32: Zadeh’s general fuzzy syllogistic scheme.

1.1.5.2 Theory of Fuzzy Syllogism

Zadeh defines fuzzy syllogisms as “an inference scheme in which the major premise, theminor premise and the conclusion are propositions containing fuzzy quantifiers” [93]. Thus,he preserves the same basic Aristotelian schema as Aristotle but expands the set of quantifiersthat can be managed considering proportional fuzzy ones. A typical example of Zadeh’s fuzzysyllogism is shown in Table 1.31. The quantifier of the conclusion, Most2 := Most⊗Most, iscalculated from the quantifiers in the premises using the fuzzy arithmetic product.

Table 1.32 shows a general schema for fuzzy syllogism, where Q1, Q2 and Q are propor-tional fuzzy quantifiers (“few”, “most”, “many”, etc.) and A, B, C, D, E and F are interrelatedfuzzy properties or terms. All the valid fuzzy syllogistic inferences are built adding differentconstraints to this schema. For instance, the example of Table 1.31 assumes that C = A∩B;E = A and F = B∩D.

R. Yager [88] extends Zadeh’s framework incorporating immediate inferences; i.e., syl-logisms with a single premise. Inferences of this kind are mainly based on the property ofmonotonicity. Table 1.33 summarizes them; the usual interpretation for ≥ Qi being at least

and for ≤ Qi at most.The reasoning process, as has already been said, is of a heuristic type for calculating a

compatible conclusion rather than a logical test for checking the logic validity of the conclu-sion. This idea is developed through the Quantifier Extension Principle (QEP) [93]. Thus, letP1, . . . ,Pn be a collection of fuzzy statements which define an absolute or proportional char-

In fuzzy logic, the extension of IA to [0,1] is the membership function, µA. Therefore, the extension of the cardinalitymeasure is

∑−Count(A) = ∑e∈E

µA(e)


Quantifier InferenceQ is monotone non-decreasing i f x > y⇒ Qx ≥ QyQ is monotone non-increasing i f x > y⇒ Qx ≤ Qy

Table 1.33: Yager’s immediate inferences.

(a) Schema

Q1 As are BsQ2 As and Bs are CsQ As are Bs and Cs

(b) Example

Most students are blondMost blond students are tallMost2 students are blond and tall

Table 1.34: Intersecion/Product syllogism.

acterization of a cardinality Pi : Ci is Qi; where Ci is the cardinality of a fuzzy set and Qi

is a quantifier. The QEP establishes that if C = f (P1;P2; ...;Pn) then Q = φ f (Q1;Q2; ...;Qn),where C is the conclusion, P1;P2; ...;Pn are the premises, f is a function, Q is the quantifierof the conclusion, Q1;Q2; ...;Qn are the quantifiers of the premises and φ f is an extensionof f by the extension principle. The main idea of the QEP is, simply, the application of theextension principle to f , obtaining a φ f that can be applied to the fuzzy numbers defined bythe quantifiers. Since the quantifiers are interpreted as fuzzy numbers, the operations betweenthe quantifiers must be fuzzy arithmetic operations.

Now we analyse each of the syllogistic inference patterns proposed by Zadeh that emergefrom the general pattern (Table 1.32).

Intersection/product is a symmetric syllogism and it is assumed C = A∩B,E = A,F =

B∩D [93]. Table 1.34(a) shows the schema, where Q1 denotes the quantifier of the firstpremise, Q2 the quantifier of the second one and Q the quantifier of the conclusion; in whichQ = Q1⊗Q2, ⊗ being a fuzzy product. Its graphic representation is shown in Figure 1.20.Table 1.34(b) shows an example10.

Multiplicative chaining (MC) is an asymmetric syllogism and it is assumed that C =

B,D =C,E = A,F =C. Table 1.35(a) shows its linguistic expression [93], where Q1 denotesthe quantifier of the first premise, Q2 the quantifier of the second premise and Q the quantifierof the conclusion; in which Q ≥ Q1⊗Q2. Table 1.35(b) shows an example and Figure 1.21its graphic representation.

10For details of the proof of all these schemas, reader is referred to [93].


Figure 1.20: Graphic representation of Intersection/product Syllogism.

(a) Schema

Q1 As are Bs (all Bs are As)Q2 Bs are CsQ As are Cs

(b) Example

Most American cars are bigMost big cars are expensiveMost2 American cars are expensive

Table 1.35: Multiplicative Chaining syllogism.

Figure 1.21: Graphic representation of Multiplicative Chaining syllogism.

The three terms involved have the following roles: A is the subject of the conclusion, C

its predicate and B is the chaining between A and C. Since the terms of the conclusion are“chained” by a middle term, this pattern is called a chaining pattern.

It is worth noting that there is an additional constraint between A and B, namely is, B⊆ A,i.e. µB (ui)≤ µA (ui) ,ui ∈U, i = 1, . . . It also can be expressed as the quantified statement all


(a) Schema

Q1 Bs are AsQ2 Bs are CsQ As are Cs

(b) Example

Most big cars are AmericanMost big cars are expensive≥ 0∨ (2 most1) American cars are expensive

Table 1.36: Major Premise Reversibility Chaining syllogism.

Figure 1.22: Graphic representation of MPR chaining syllogism.

B are A. This constraint is relevant as it allows us to ascertain the distribution of the elementsin sets A and B. Without this information, the conclusion cannot be calculated.

Taking all the previous considerations into account, the procedure for calculating the con-clusion is shown in Equation (1.25), where ⊗ denotes a fuzzy product and thus the chainingpattern is called multiplicative.

Q≥ (Q1⊗Q2). (1.25)

Major premise reversibility chaining (MPR) is an asymmetric syllogism and it is as-sumed that C = B,D =C,E = A,F =C. The schema thereof is shown in Table 1.36(a), whereQ1 denotes the quantifier of the first premise, Q2 the quantifier of the second premise and Q

the quantifier of the conclusion. Table 1.36(b) shows an example and Figure 1.22 its graphicrepresentation.

We can consider this model as a variant of the MC pattern, where the constraint B ⊆ A

is substituted by the reversibility of the first premise; i.e., Q1 A are B↔ Q1 B are A. Zadehpoints out that this semantic equivalence is approximate rather than exact, but the calculationof how “approximate” it can be remains a non-trivial open question [93].


(a) A. Conjunction

Q1 As are CsQ2 Bs are CsQ As and Bs are Cs

(b) A. Disjunction

Q1 As are CsQ2 Bs are CsQ As or Bs are Cs

Table 1.37: Antecedent Conjunction/Disjunction schemas.

(a) A. Conjunction

Many students are tallMost teachers are tall[None,all] teachers and students are tall

(b) A. Disjunction

Many students are tallMost teachers are tallMany students or teachers are tall

Table 1.38: Antecedent Conjunction/Disjunction examples.

Nevertheless, there is a notable obstacle for this constraint. For proportional quantifiers,it does not hold that a quantified sentence (e.g., “most American cars are big”) and anothersentence with the arguments interchanged (“most big cars are American”) are semanticallyequivalent. However, it is true when we are talking about absolute quantifiers; e.g., “arounda hundred thousand American cars are big” is semantically equivalent to “around a hundredthousand big cars are American”.

The procedure for calculating the conclusion, shown in Equation (1.26), involves fuzzyaddition (⊕) and subtraction () arithmetic operations.

Q≥ max(0,Q1⊕Q21) (1.26)

Antecedent Conjunction/Disjunction is a symmetric syllogism and it is assumed thatB = C,C = B,D = C,F = C. The subject of the conclusion for the Antecedent Conjunction(AC) pattern is constituted by two conjuncted terms assuming that E = A∩B (schema shownin Table 1.37(a)); the subject of the conclusion for the Antecedent Disjunction (AD) pattern isconstituted by two disjuncted terms assuming that E =A∪B (schema shown in Table 1.37(b)).Table 1.38 shows an example of each pattern and Figure 1.23 their graphical representation.

The AC pattern entails assuming that the items of evidence are conditionally indepen-dent [93]. This fact leads us to dismiss it from the perspective of syllogistic reasoning as thistype of additional hypothesis is beyond the scope of the present research.

The AD pattern does not need additional constraints. The conclusion is Q ≤ (Q1⊕Q2),⊕ being a fuzzy addition.


(a) A. Conjuntion

OR

(b) A. Disjunction

Figure 1.23: Graphic representation of A. Conjunction and A. Disjunction.

Having described Fuzzy Syllogistics, we select a number of critical reviews. In [32], E.Kerre and Y. Liu, shows the disadvantages of QEP. Firstly, different results can be obtained,depending on how the cardinality equations are written; in consequence, ambiguity arises inthe result of a syllogism11. Moreover, inference schemes are not reversed either with respectto addition or multiplication. Finally, there is an incompleteness of fuzzy quantifiers and fuzzynumbers, and, therefore, a good chance of getting a discontinuous result [32].

1.1.5.3 Summary

Zadeh’s model was the first that dealt with a genuine fuzzy syllogism; i.e., the vague quan-tifiers of the corresponding premises are represented through fuzzy sets. Furthermore, somenew schemas of inference are proposed, both asymmetric and symmetric ones. We shall nowgo on to check the compatibility of Aristotelian Syllogistics with Zadeh’s framework [50, 51].

Firstly, it is worth noting that no symmetric syllogisms are valid to manage Aristoteliansyllogisms as they are not supported on the chaining of terms. Furthermore, the intersec-tion/product scheme involves logic operations in the second premise and the conclusion thatdo not appear in the classical moods. Therefore, none of these patterns can be considered forthis analysis. Thus, only the MC and MPR patterns are possible candidates.

11For instance, in the example of Table 1.31, if most = 34 , the quantifier of C is

( 34

)2. This functional relationship

can also be written as most3

most , but in the case of fuzzy arithmetic with fuzzy numbers the equivalence between most2

and most3

most is not fulfilled.


Pattern Major premise Minor premiseFigure I Subject PredicateFigure II Predicate PredicateFigure III Subject SubjectFigure IV Predicate SubjectMC Subject PredicateMPR Subject Predicate

Table 1.39: The position of the middle term in Aristotelian Figures and Zadeh’s MC and MPRpatterns

AAA EAE AII EIO AAI EAOMC No No No No No NoMPR No No Yes No No No

Table 1.40: Behavior of Zadeh’s patterns with respect to Figure I.

If we consider their capability to reproduce the four classical figures, only Figure I iscompatible, as shown in Table 1.39, owing to the position of the middle term. Therefore,Figures II, III and IV cannot be modelled using this approach. This limits the scope of themodel and hence only the six Aristotelian moods of Figure I can be checked (see Table 1.40).Our result is that only the AII mood is compatible with the MPR scheme (i.e., one mood outof twelve).

The problem in the MC pattern is in the constraint B ⊆ A in the first premise (minorpremise in Aristotelian terms), which is too restrictive. In this set of Aristotelian moods, theconverse of the minor premise in linguistic terms is some B are A; that is, A∪B 6= /0; forinstance, in the AII mood, B ⊆ A cannot be inferred from the second premise, I: some A are

B. This condition is less restrictive than Zadeh’s and thus many of the possible inferencesdismissed by Zadeh’s pattern can be solved using the Aristotelian framework.

The MPR pattern is only valid for the AII mood, since the statement and its conversein the minor premise have the same quantifier (“some”). For the other moods, those thatinclude A in the minor premise present the same problem as in the MC pattern. The solutionof the EIO mood, which is compatible, presents a problem: the value obtained for Q is 0instead of [0,1− ε) with ε ∈ (0,1], the usual representation for the quantifier “not all” ina proportional interpretation. The origin of this problem is that Zadeh’s framework is notcapable of managing moods involving decreasing quantifiers, such as “not all”. In conclusion,


Zadeh’s approach to syllogistic reasoning is compatible with the AII mood of AristotelianSyllogistics, but only for the MPR scheme.

On the other hand, as stated in [32], Fuzzy Syllogistics also shows relevant problems inthe QEP, the core of its reasoning process, and from this point of view, it is a weak model.However, it introduces an interesting matter not considered until now: additional constraintsor premises that are not necessarily explicit in the argument. For instance, MC and MPRmodels need assumptions that are not considered as premises in the reasoning schemas; whatis more, if they are not included in the syllogism, it is not valid. In this case, we can think thatZadeh is assuming that they belong to the background or the context shared by the users and,therefore, they need to be clearly established to avoid fallacies in reasoning.

1.1.6 Generalized Intermediate Syllogistics

Another proposal that manages syllogistic arguments involving fuzzy quantifiers is the onedeveloped by V. Nóvak and P. Murinová [42, 37] which aims to generalize Intermediate Syl-logistics (see section 1.1.2) called, for that reason, Generalized Intermediate Syllogistics.

It is developed in the framework of fuzzy type theory (FTT), a higher order logic, wherethe intermediate quantifiers are introduced as special formulae using the formal theory oftrichotomous evaluative linguistic (TEv) [41]. According to [42], this framework presents thefollowing advantages:

1. The standard syntax of FTT is sufficient for managing intermediate quantifiers as theyare only short forms for special formulas.

2. This theory is general enough to cover a broad class of intermediate quantifiers andprovides a unique definition for them.

3. Syntactical properties of the quantifiers can be studied separately from their interpreta-tion (semantics). Therefore, syllogisms can also be studied from a syntactical point ofview, independently of their semantics.

We shall focus on a intuitive description of the main concepts of the framework, as wellas on their main contributions to the study of syllogism. The formal development of thisframework can be found in [37, 41, 42].

This section is organized as follows: section 1.1.6.1 describes how quantified statementsare formalized; section 1.1.6.2 explains the new tool for interpreting quantifiers; section 1.1.6.3


describes a generalized version of the LSO; section 1.1.6.4 describes the main characteristicsof the reasoning process proposed in this framework; and, finally, section 1.1.6.5 summarizesthe strengths and the weakness of this proposal.

1.1.6.1 Theory of Generalized Intermediate Categorical Statements

Generalized Intermediate Categorical Statements are explicitly introduced as a generalizationof Peterson’s intermediate quantifiers (see section 1.1.2). Thus, they have the typical structureof quantified statements (Q S are P) and the same type of quantifiers (proportional ones). Themost relevant contribution is the development of a formal theory for managing it.

Accordingly, quantified statements are formalized through FTT. Classical Type Theory iswell known in mathematics and logic. It is a higher order logic built as an extension of first-order logic. It was proposed by A. Whitehead and B. Russell in their renowned book Principia

Mathematica [87] published in three volumes between 1910 and 1913. The concept of type

was introduced to overcome the Russell paradox12, also known as barber paradox, and, inthis way, defends the thesis of logicism; that is, mathematics is reducible to logic. Thus, type

is a new kind of category that allows us to form a hierarchy and assigns each mathematicalentity to a type. For instance, a natural number is a type. As in classical logic, the structureof its truth-values is {0,1}.

FTT is an extension of classical type theory to fuzzy logic. Therefore, it is still a higherorder logic but the structure of the truth-values is [0,1] [40]. It is relevant to note that, althoughthe motivation of classical type theory is mathematics, the motivation of FTT is linguistics andthe management of natural language.

Aristotelian quantifiers are also included in this framework, through their equivalent defi-nition in first order logic; that is13,

all S are P≡ ∀x(Sx⇒ Px)

some S are P≡ ∃x(Sx∧Px)(1.27)

and their corresponding negations;

no S are P≡ ∀x(Sx⇒¬Px)

some S are not-P≡ ∃x(Sx∧¬Px)(1.28)

12Russell wrote it to G. Frege in a letter on June 16 [69] meanwhile he was working on Principia Mathematica.13In type theory,⇒ is equivalent to→.


Thus, any quantified statement in the standard form Q S are P, whose quantifier is mod-elled through TEv, is expressed as a special formula in the FTT attending to its positionbetween the universal and the particular quantifiers.

1.1.6.2 Trichotomous Evaluative Linguistic (TEv)

Many of the expressions human beings use in their daily life are expressions such us “tall”,“very tall”, “medium”, etc. These expressions belong to the class of evaluative linguistic

expressions, which “typically characterize a person’s behaviour or attitude in terms of thespeaker’s subjective judgement” [31].

A subclass of evaluation linguistic expressions comprises trichotomous evaluative expres-

sions, which are usually constituted by three terms: a nominal adjective, its antonym and amedium one between them. Its canonical form is of the type small-medium-big. They arefrequently used to evaluate, judge, estimate, qualify, etc. different kind of situations. TEvexpressions have the following properties:

– They are adjectives of manner; i.e, the are subjective and context dependent.

– They constitute a bounded ordered scale. Such a scale can consist of real measuringunits (centimetres, litres, etc.) or some kind of abstract units (happiness, sadness, etc.).

– They are essentially vague.

As a vague part of natural language, they were dealt with by fuzzy logic. However, inmost cases, they are interpreted by simple fuzzy sets in the universe of real numbers, usingtriangular or trapezoidal membership functions. Although fuzzy logic is sensitive to the con-text, this type of definition does not adequately manage its scalar character; i.e., the meaningof “small” is not only a question of centimetres or meters but what the definition of “big” is.

Nóvak [41] provides a formal mathematical model for dealing with this complex andvague meaning using FTT. He assumes the following hypothesis [41, p.2941]:

Vagueness of the meaning of natural language expressions is a consequence ofthe indiscernibility phenomenon. Therefore, any extension of a natural languageexpression can always be characterized by the same specific indiscernibility rela-tion.


Thus, the semantics of the indiscernibility TEv expressions is characterized in FTT throughthe use of fuzzy equality, an imprecise equality that characterizes the similarity between twoobjects. This idea has been widely studied in the fuzzy logic literature and its analysis isbeyond the scope of this research.

Nóvak interprets intermediate quantifiers as a kind of TEv, the universal and the exis-tential quantifiers being the extremes and any other intermediate quantifier between them,the medium element of the scale. The extremes, in this case, have crisp definitions and themedium corresponds with a fuzzy set defined as a membership function. Hence, any inter-mediate categorical statement is defined by one of the following formulas14, where x and z

denote any element of the universe of the discourse U and S and P the standard term-sets:

(Q∗∀Evx(S,P))≡ (∃z)((z⊆ S)∧Ev(µB)z∧ (∀x)(zx⇒ Ax)) (1.29)

(Q∗∃Evx(S,P)) ≡ (∃z)((z⊆ S)∧Ev(µB)z∧ (∃x)(zx∧Ax)) (1.30)

As we can see, this definition consists of three parts:

(∃z)(z⊆ S) It is the existential import, that is, the subject term of intermediate categoricalstatements cannot be an empty set. However, it is relevant to note that the existentialimport is not inherent to his definition but it is explicitly added; on the other hand, italso shows the left edge of the scale because it is equivalent to say that “there is at leastone element in S”.

Ev(µB)z This represents the evaluative process that generates the fuzzy set corresponding tothe meaning of the linguistic quantifier such as “most”, “many”, “few”, etc. It corre-sponds with the medium member of the scale.

(∀x)(zx⇒ Ax)/(∃x)(zx∧Ax) It is the right-hand member of the scale.

It is relevant to note the syntactical character of this definition of the quantifiers. Althoughthe fuzzy set that defines the meaning of the quantifier changes, the general formula is stillvalid. It should also be noted that the TEv definition substitutes the role of LSO in this frame-work, as it defines the meaning of the intermediate quantifiers according to the dependenceswith the universal and particular ones.

Regarding the quantifiers considered by Nóvak, he directly refers to the analysis providedby Peterson (see section 1.1.2; 5-quantity LSO).

14Here, we avoid the use of the specific formalism of FTT as it has not been adequately defined. At pointed out,the main aim of this section is to present the general idea of this approach rather than its technical details.


1.1.6.3 Generalized Logic Square of Opposition with Intermediate Quantifiers

In [36], P. Murinová and V. Novák develop a generalized version of the LSO proposed by P.L. Peterson and B. E. Thompson (see section 1.1.2.1, Figure 1.7) in the framework of FTT.We stated that, in all but Zadeh’s Syllogistics, the role of an LSO is fundamental. In IntervalSyllogistics, when the authors propose a linguistic version of their model, it is substituted bya biordered structure different from a LSO but which plays an analogous role. In conclusion,any syllogistic framework that manages linguistic quantifiers needs a version of the LSO.

P. Murinová and V. Novák assume that “intermediate quantifiers are classical general (∀)or existential (∃) quantifiers, but the universe of quantification is modified, and the modifi-cation can be imprecise” [36, p. 12]. Using a FTT that extends the TEv, they can managethem under a fuzzy framework. They obtain a high general and robust result as they only usesyntactical proofs, which also involve the Aristotelian LSO as a particular case.

Regarding the problem of existential import, they presuppose that any term-set involvedin a quantified statement are non-empty sets. They use first-order logic in the formulation ofuniversal and include the existence of S by conjunction into the Aristotelian proposition; forinstance, A := (∀x)(Sx⇒ Px)&(∃x)x, where & denotes the corresponding fuzzy conjunction.With this assumption, all the relationships of an LSO are proved. Figure 1.24 shows thesquare obtained, where a,e, p,b, t,d,k,g, i and o denote the truth value for each statement, nota fuzzy number associated to the quantifier, as P. Murinová and V. Novák only consider logicdefinitions of them. Thus, for example, if “almost all students are tall” with p = 0.4 then“most students are tall” is with g = 0.45 since p≤ t15.

There are some relationships from Peterson and Thompson’s 5-quantity LSO that are notincluded in this version; specifically, the relations between the statements K, O, G and I areomitted. The reason is that K and G have undefined meanings and the relationships betweeneach one and with other quantifiers are much too complicated. The authors point out thatdealing with these quantifiers will form part of subsequent investigations.

1.1.6.4 Theory of Generalized Intermediate Syllogism

The main innovation of this framework is the independence of syntax and semantics and howthe role of the LSO is avoided using TEv. In this way, the validity of syllogisms is provedusing syntactical proofs, which confers a great deal of robustness to this proposal.

15The proofs can be confronted in [36]


A: a ≤ p E: e ≤ 1 - p

I: i= 1 O: o =1

P: p

K: t ≤ k

T: p ≤ t

B: e ≤ b≤ 1 - p

D: b ≤ d

G: d ≤ g

Figure 1.24: Generalized 5-Quantity LSO.

Nóvak distinguishes between valid implications and valid syllogisms [42, p.1240]. Theformer correspond to the concept of immediate inferences but they are exclusively based onthe property of monotonicity; i.e., “wider” quantifiers entails “narrower” ones. The examplesare well known, for instance, “almost all students are tall” implies “most students are tall”16.The latter correspond with the standard form of syllogism (three terms, two premises and aconclusion) but adopting a formulation in terms of propositional logic; i.e., P1 stands for theMP, P2 denotes the NP and C stands for the conclusion. Thus, a syllogism is formally valid ifP1&P2⇒C.

In [42, 37] all the valid intermediate syllogisms are proved and all the invalid ones aredischarged. Hence, the compatibility between both approaches and the generalization to fuzzylogic of this framework respect to the previous one is proved.

1.1.6.5 Summary

Generalized Intermediate Syllogistics is a generalization of Intermediate Syllogistics in fuzzylogic. It is relevant to note that it is the only fuzzy approach that explicitly considers Aris-totelian Syllogistics as a particular case. Furthermore, it also analyses syntax and semanticsindependently obtaining, by this way, such a strong model.

On the other hand, the TEv framework is a genuine fuzzy proposal for interpreting in-termediate quantifiers without using an LSO. However, from the point of view of inference

16The proof is explained in [42, p.1240].


schemas, Nóvak only considers proportional quantifiers and arguments with the typical form:three terms, two premises and a conclusion.

1.2 Conditional Interpretation

Conditional interpretation assumes that the deep structure of quantified statements is condi-tional; i.e., an implication where a relationship between an antecedent and a consequent isestablished. For instance, “all natural numbers are real numbers” ≡ “if something is a naturalnumber, then it is necessarily a real number”, where “necessarily” stands for the quantifier“all” and determines the power of the link between the antecedent part and the consequentone. This interpretation entails a different categorization of the elements that compound aquantified statement:

– Subject and predicate are not sets but propositions. So, instead of talking about aboutsubject-term and predicate-term, we have to talk in terms of antecedent or sufficientcondition (which corresponds with the subject-term) and consequent or necessary con-dition (which corresponds with the predicate-term) of the conditional. For instance,the subject-term natural number is transformed into the proposition “if something is anatural number”.

– Quantifier does not denote a quantity relationship between two terms but how strong isthe link between the antecedent and the consequent of the implication. For instance, thequantifier “all” indicates a perfect link between if-then; i.e., “if something is a naturalnumber then it is necessarily a real number”; other quantifiers such as “many humanbeings are Chinese” are equivalent to “if something is a human being then it is quite

probably Chinese”.

– The universe of discourse is not restricted to a determined set rather it is unconditional.Thus, “if something is a natural number then. . . ” refers to any element of the uncondi-tional universe of discourse endowed with the property of being a natural number; notspecifically the set whose elements are the natural numbers.

From these differences some additional conclusions can be inferred. Firstly, set-basedinterpretation allow us to interchange the subject-term and the predicate-term in a categoricalstatement and infer a new valid statement, but it cannot deal with singular terms (those thatrefers to a concrete being, not to sets); nevertheless, conditional reading does not allow us that

1.2. Conditional Interpretation 81

direct conversion in the statements although it can manage propositions that involve singularterms [29] in a straightforward manner. For instance, from “all human beings are mortal” wecan infer “some mortal beings are human beings”17; however, from a statement such as “GaiusIulius Caesar is a human being” the same kind of inference cannot be done because Gaius

Iulius Caesar refers to a concrete historical person and not to a set. Using the conditionalinterpretation, the equivalent statement to “all human beings are mortal” is “if something isa human being then it is mortal”, where something refers to any element of the universe andhuman being and mortal being refers to the properties being human and mortal respectively.The interchange between both terms produces the statement “if something is mortal then itis a human being”, that is not true and, then, the interchange is not acceptable; however, theconditional interpretation can deal perfectly with sentences that involve singular terms suchas “Gaius Ilius Caesar”; i.e., “if something is Gaius Iulius Caesar then it is a human being”.

Another corollary deals with the so-called problem of existential import [29]. The bestway to illustrate this is through a question. Let us consider the statement “all gnomes areshort”. Is it true or false? First of all, we agree that gnomes do not exist. From a set-basedinterpretation and assuming the existential import, it can only be true if and only if thereare gnomes; therefore, it is false. Without assuming the existential import, the statement isvacuously true but true, because gnomes denotes an empty set ( /0), that is a subset of anyset. In this case, some inferences of the LSO are not valid, such as subalternation; for thisreason, in general, all syllogistic approaches based on the set-based interpretation assume theexistential import. However, for the conditional reading this is an irrelevant question becausethe inferences of the LSO is a type of inference not considered.

From the point of view of Linguistics, the set-based interpretation is compatible with theTGQ [86] and this allows us to introduce a new kind of linguistic quantifier, not tradition-ally considered in the framework of syllogistic reasoning, which increases the number ofarguments that can be managed. Nevertheless, although proportional quantifiers have a clearconditional reading, the interpretation of other kinds of quantifiers as absolute (“twenty five”,“three”, etc.), exception (“all but three”, etc.) or comparative (“double”, etc.) is not so clear.For instance, the meaning of “all students but two are blond” is clear in the set-based reading,but how can it be expressed using the conditional reading? It can be something like if x is a

student then x is blond [0.98]? Why not [0.99]?

17This kind of inference is detailed explained in section 1.1.1.


If we consider now the pragmatic dimension of language, there are also a number of ad-ditional considerations that must be discussed. Quantified statements, as a type of categoricalstatements, are defined as assertions; i.e., statements that can be true or false because theydescribe a particular fact or event of the reality. For instance, “all natural numbers are realnumbers” describes a characteristic of natural numbers. This statement has neither uncer-tainty nor vagueness. However, its equivalent conditional statement, if x is a natural number,

then x is a real number, is not an assertion; rather it describes a condition for the possibilityof an event (choose a x that, in the case of having the property natural number, it also has theproperty of real number), with the corresponding uncertainty associated to it. For this reason,it is assumed that the conditional reading is a kind of probabilistic reading that can be equiv-alent to the set-based reading from an operative point of view, but they are not completelyequivalent in other dimensions of natural language.

This section is organized as follows: section 1.2.1 describes the framework proposed byM. Oaksford and N. Chater in the field of psychology of reasoning; section 1.2.2 contains theframework proposed by M. Spies based on Dempster-Shafer theory and, finally, section 1.2.3introduce briefly the proposal of syllogism by D. Schwartz.

1.2.1 Probabilistic Syllogistics

M. Oaksford and N. Chater [43] propose a framework to deal with syllogistic reasoning interms of probability, also providing a set of “fast and frugal” heuristics instead of the usuallogical and deductive framework. They assume the following hypothesis:

– In their daily life, human beings manage incomplete information, uncertainty and vagueconcepts all the time. Furthermore, our main way of communication, natural language,involves many of this kind of concepts, and this is not inconvenient for making infer-ences. Therefore, human ordinary reasoning must be managed using tools that are ableto deal with imprecision and uncertainty.

– Probability and logic deal with disjoint sets of phenomena of human reasoning; i.e.,there is no competition between them but they are complementary. However, if genuinelogical tasks can be dealt with using a probabilistic approach, it is possible to concludethat a general theory of human reasoning must be a probabilistic framework.


– Aristotelian Syllogistics is the core area of logical deductive reasoning. Therefore,modelling syllogistic reasoning using a probabilistic framework is a significant contri-bution to a probabilistic general theory of human reasoning.

The main objectives to be achieved are:

– The authors work in the field of psychology of reasoning, not in logic (the formerattempting to understand how human beings make inferences and the latter trying todiscover the rules of the right reasoning, independently of how human beings executethem). As a consequence, the psychology of reasoning studies any reasoning processthrough which people execute an inference (valid or not) while logic focuses strictly onthe validity of arguments.

– The authors’ main aim is to describe the set of procedures that human beings use in thereasoning. In this way, they interpret reasoning in terms of heuristic strategies used bythe reasoner to achieve a conclusion accepted as true, even though a subsequent logicalanalysis shows that it is invalid.

– The evidence of this framework is the psychological experiments with individualsanalysing whether the conclusion predicted by the model and the conclusion achievedby the person match. Logical proofs of the arguments are not a part of the experiments.

We shall focus mainly on the interesting ideas from the point of view of this research andon the criticism made by B. Geurts [23] about the management of non-proportional quanti-fiers.

This section is organized as follows: section 1.2.1.1 describes how quantified statementsare interpreted in terms of probability; section 1.2.1.2 describes the inference mechanism;section 1.2.1.3 explains the criticism formulated by B. Geurts of this probabilistic model’scapabilities for representing other kinds of quantifiers, and, finally, section 1.2.1.4 summarizesthe main characteristics of this framework.

1.2.1.1 Theory of Probabilistic Quantified Statements

Oaksford and Chater start with the following assumption regarding quantified statements [43,p. 219]:

Quantified statements can be given intuitively natural meanings in terms of con-straints on the conditional probability of the predicate given the subject.


Linguistic Expression Probabilistic ExpressionAll S are P P(P|S) = 1No S are P P(P|S) = 0Some S are P P(P|S)> 0 and S P are not emptySome S are not P P(P|S)< 1 and S not−P are not emptyMost S are P 1− ε ≤ P(P|S)< 1,ε > 0Few S are P 0 < P(P|S)≤ ε,ε > 0

Table 1.41: Probabilistic expression of “all”, “none”, “some”, “some. . . not”, “few” and“most”.

Thus, a quantified statement such as all S are P means that the probability of P given S is1; in probability notation, P(P|S) = 1. For intermediate quantifiers, such as “most”, “many”,etc. the value of the probabilities changes in the interval [0,1]. For instance, most S are P

means that the probability of P given S is less than 1 but high; i.e., 1−ε ≤ P(P|S)< 1, whereε > 0 although ε changes according to meaning of the linguistic quantifier. Thus, for few S

are P the probability writing is 0 < P(P|S)≤ ε , where ε is small as it represents the meaningof “few”. Table 1.41 shows the probabilistic definition for the Aristotelian quantifiers and“most” and “few”. It is relevant to note “all” does not include the existential import.

A new concept not explicitly used previously in the analysis of quantification propositionsis introduced: the informativeness (I) of a statement (s), I(s). It is taken from informationtheory developed by Shannon and Waver [73], where I(s) is inversely related to its probability(P(s)):

I(s) = log2

[1

P(s)

](1.31)

Thus, given two quantified statements, B and Γ, if B→ Γ then I(Γ)< I(B). This providesa criterion for establishing a partial order among the quantified statements:

I(All)> I(Some)

I(Most)> I(Some)

I(Few)> I(Some)

I(Most)> I(Some . . .not)

I(Few)> I(Some . . .not)

I(None)> I(Some . . .not)

(1.32)


This partial order can be transformed into a total order with an additional assumption [43,p. 221]: the properties referred to in natural language typically apply to a small proportionof possible objects. Authors qualify it as a ‘rarity’ attending to the frequency of quantifiedstatements that are true and, hence, informative. Many of the predicates that human beingsmanage are mutually exclusive, for instance cat and dog: no being can be a cat and a dogsimultaneously; in other words, no dog is a cat. In general, “no” statements are true and notso informative because they describe information that is usually assumed as hidden premisesin our arguments. It is true that at other times, “no” statements can be very unexpected andhighly informative as they describe surprising relationships, but this is not the most commonsituation. “Some. . . not” sentences are even less informative than “no” statements as they canbe predicated of almost all properties expressed in natural language; they include not only“no” statements but also “some” statements.

I(All)> I(Most)> I(Few)> I(Some)> I(None)>> I(Some . . .not) (1.33)

Expression (1.33) is the generalization of (1.32). It shows the total order I with this as-sumption, where > stands for “more informative than” and >> stands for “much more infor-mative than”.

1.2.1.2 Theory of Probabilistic Syllogism

Probabilistic Syllogistics preserves the canonical form of the syllogisms: arguments com-posed by three terms, two premises and a conclusion relating two end terms (NT and MT )via a middle term (DT ). Hence, the premises express the conditional probabilities amongNT , MT and DT ; the conclusion, qualified as p-valid18, can be calculated if the constraintsexpressed in the premises are sufficient for calculating the probability of MT given the NT .The general schema for the four Aristotelian Figures is shown in Table 1.42, where α andβ stand for the probability defined by the quantifiers and f (α,β ) denotes the correspondingprocedure for calculating the conditional probability between the end terms. It is worth notingthat this framework does not distinguish between MP and NP but between max-premise andmin-premise, the first one being the most informative and the second one the least informative.

As it pointed out previously, Oaskford and Chater do not focus principally on determininghow the f of f (α,β ) must be, since it can be calculated using probabilistic logic [45], but

18As has already been pointed out, this probabilistic framework is focused on a notion of validity different to theset-based approaches. Hence, the authors develop a specific concept of validity applied to probabilistic statementsknown as probabilistically-validity of p-validity.


Figure I Figure IIP(MT |DT ) = α P(DT |MT ) = α

P(DT |NT ) = β P(DT |NT ) = β

——————– ——————–P(MT |NT ) = f (α,β ) P(MT |NT ) = f (α,β )

Figure III Figure IVP(MT |DT ) = α P(DT |MT ) = α

P(NT |DT ) = β P(NT |DT ) = β

——————– ——————–P(MT |NT ) = f (α,β ) P(MT |NT ) = f (α,β )

Table 1.42: Inference schema of syllogism under a probabilistic approach.

on the human strategies for calculating the conclusion of the syllogism; i.e., the heuristics toachieve h(α,β ), that can match or not with f (α,β ).

The probability heuristics model tries to deal with the ‘fast and frugal’ way of human be-ings for inferring conclusions. It is divided into two steps. The first being to order the premisesaccording to their informativeness following (1.33). Once the premises are ordered, the sec-ond step is to apply the heuristic method to obtain the conclusion. The authors distinguishfive methods [43, p. 217-219]:

min-heuristic The quantifier of the conclusion is the same one of the least informative prem-ise. It is the most preferred method.

P-entailments It means probabilistically-entail and address, more or less, the subalternationof LSO. For instance, if the conclusion of a syllogism is “all crows are black”, then thep-entail is “some crow is black”. It is the second most preferred conclusion.

attachment-heuristic It deals with the order of the terms in the conclusion. It states that thereis a tendency towards selection the subject of one of the premises as the subject of theconclusion.

max-heuristic It is applied to choose the most informative conclusion. It predicts that thereshould be a linear order in the frequency of the min-heuristic response dependent on themax-premise such that “all” > “some” > “some. . . not” > “none”.


Syllogism HeuristicAll DT are MT max-premiseSome NT are DT min-premiseSome-type conclusion by min-heuristicSome NT and MT by attachment-heuristic

Table 1.43: Inference schema of syllogism AII (Figure I) under a probabilistic approach.

some. . . not-heuristic It is applied to avoid producing or accepting “some. . . not” conclu-sions, as they are the least informative. Therefore, syllogisms with “all”, “some”, or“none” conclusions are more acceptable that those with “some. . . not” conclusions.

The first three methods are oriented toward generating the conclusions of syllogisms,while the last two are procedures for determining the reliability or confidence of the con-clusion achieved. The authors illustrate the procedure with the AII mood of Figure I shownin Table 1.43.

1.2.1.3 The Case of Non-Proportional Quantifiers

As has been shown until now, Oaksford and Chater introduce new quantifiers defined as prob-abilities, but considering exclusively proportional ones (i.e.; “most” and ”few”). Other quan-tifiers, such as absolute (“five”, “twenty-five”, etc.) or exception ones (“all but three”, “all butfive”, etc.) are not explicitly dealt with.

B. Geurts [23] points out precisely this question as a weakness of this system, as TGQinvolves more quantifiers than the proportional ones. The starting point is the difficulty ininterpreting an absolute quantifier in a probabilistic interpretation of binary quantified state-ments; for instance, what is the probabilistic interpretation of “five students are tall”?

Answering this question, Oaksford and Chater dismiss the introduction of non-proportion-al quantifiers saying that:

1. TGQ is based on a second order logic. Second-order logics, in general, are not com-pletely decidable and, therefore, there are propositions that cannot be proved or rejected.There is an exception with monadic second-order logic, which is decidable but it canonly express propositions composed by a single argument such as “Sherlock Holmesis a consulting detective”. However, most of the quantified statements that human be-


ing manage are binary; i.e., “few people are consulting detectives” and some of themcannot be decidable.

2. A direct relationship between proportional and absolute quantifiers can be establishedif the size or the cardinality of the subject is known. For instance, let us consider a setof students with |students| = 10, “two students are tall” can be easily read as “somestudents are tall”, “few students are tall” or “almost all students are not tall”. For thatreasoning, the effort of introducing absolute quantifiers in a probabilistic framework isnot worthwhile.

3. The use of precise absolute quantifiers is very reduced in our daily use of natural lan-guage as they only appear in very specific contexts. Furthermore, precise assertions donot generate useful inferences if people can make them using proportional quantifiers.

4. Geurts’ critique of probabilistic heuristic model is supported by formal arguments. Hedoes not offer the corresponding psychological experiments to justify his arguments.

To the aim of this research, the contra-arguments explained by Oaksford and Chater arerelevant and must be analysed in detail. Hence;

1. It is true that no monadic second-order logic is not completely decidable. However,there are many statements that are decidable and, very likely, the formalization of mostof quantified statements that we use in our daily life belong to this class. Hence, TGQcannot be disregarded as an adequate tool for the analysis of syllogistic reasoning eventhough it is a second-order logic.

2. The direct relationship between proportional and absolute quantifiers demands addi-tional information that may not be available in all cases. Furthermore, it is possible tobuild very simple and interesting arguments that combine both kind of quantifiers; forinstance, “almost all students are tall and twenty-five students are blond; therefore, mostblond students are tall”. It is not difficult to build a counter-example of this syllogism(the non-tall students are the twenty-five blond students), but assuming as previous con-ditions an independence between both sets and a uniform distribution of the elements,the conclusion is acceptable.

3. The model of syllogistic reasoning that we propose in this work tries to deal with com-mon and uncommon situations. For this reason, although absolute and precise quanti-


fiers are less common than proportional ones in many contexts, we assume the possi-bility of managing them jointly as a challenge.

4. The research of the present work does not belong to the field of psychology of reasoningbut to the field of logic and computer science. Thus, the validity of a syllogistic modelmust be supported in other kind of experiments and the obtained results should not benecessarily compared with psychological experiments.

1.2.1.4 Summary

Probabilistic Syllogistics interprets quantified statements in terms of conditionals propositionswhere the quantifiers denote the strength of the conditional in terms of percentages. “All” and“no” are the bounds (100% and 0%, respectively) and quantifiers such as “most”, “few”, etc.are percentages between them. Nevertheless, the interpretation of other quantifiers defined byTGQ, such as absolute or exception ones, is not so clear in terms of probabilities.

This system emphasizes the psychological and heuristic dimension of syllogism insteadof the logical one. Aristotelian Syllogistics is deeply analysed and the compatibility betweenthem its checked, although other possible inferences schemas are not considered. Nonethe-less, an interesting contribution of this system is the relevance of the background of the indi-viduals or the implicit premises in the arguments to achieve acceptable inferences.

Although the proposal of this memory follows the formal and logic vein, there two inter-esting ideas that will be fruitful:

– The human inference process can also be coherently interpreted as a heuristic process.The premises are constraints in a searching space of possible solutions, where the con-clusion is one of the possible solutions (eventually the optimal one) and the reasoningrules are not deductive logical rules but strategies for checking the compatibility be-tween the premises and the conclusion.

– With a number of additional assumptions, it is easy to establish an equivalence betweenset-based approaches and probabilistic ones, therefore, it is possible to develop a modelof syllogistic reasoning that can also manage knowledge expressed by probabilities.


1.2.2 Support Logic Syllogistics

M. Spies [79] proposes another system of syllogistic reasoning that is known as SupportLogic Syllogistics. It is based on a conditional interpretation of quantified statements wherethe quantifier indicates the degree of evidence or belief that supports the conditional. Thisapproach is based on Dempster-Shafer’s Theory of Evidence [15, 72], a discipline that belongsto the field of formal study of beliefs; i.e., epistemology. Compared with probability, the mainpoint of theories of evidence is that the strength of a proposition depends on the knowledgeof the agent rather than on objective evidence.

The degree of belief of any quantified statement is a value that can be divided into threejointly exhaustive parts. Thus, a speaker’s knowledge or belief about a proposition Γ entailsthe following:

1. Usually, an agent has a degree of belief in the truthfulness of a proposition. For instance,considering the adventures of Sherlock Holmes, J. Watson’s degree of belief that S.Holmes has died in Reichenbachfall equals 0.8. This is the part that speaks in favour ofΓ (the proposition that express the belief of J. Watson); i.e. Bel(Γ).

2. The same agent also has a degree of belief in the truthfulness of the opposite propositionof Γ, that is W\Γ. For instance, J. Watson’s degree of belief that S. Holmes has not diedin Reichenbachfall equals 0.1. This is the part that speaks against Γ; i.e. Bel(W\Γ).

3. Finally, a part that neither speaks in favour of nor against Γ can also appear. Thus, con-sidering the previous examples, where Bel(Γ) = 0.8 and Bel(W\Γ) = 0.1, the additionis less than 1 (0.9) and, therefore, there is a neutral part, Ignorance (I) (I(Γ) = 0.1), thatmust be distributed among both parts. If the addition of Bel(Γ) and Bel(W\Γ) equals1, there is no ignorance.

Over these three concepts, a new concept known as “degree of plausibility” (P(Γ)) rises.It quantifies the part of the agent’s knowledge compatible with the the proposition Γ; i.e., theaddition of the degree of belief and ignorance in Γ obtaining P(Γ) = 1−Bel(W\Γ).

For instance, J. Watson has asserted the statement Γ. The P(Γ), J. Watson’s belief, isP(Γ) = 1−0.1 (P(Γ) = 1−(Bel(W\Γ))); that is, P(Γ) = 0.9. This value can match Bel(Γ) ornot; in this case it is not since I(Γ) = 0.1. It is worth noting that in the case of a proposition Γ

with P(Γ) = 0, then P(Γ) = Bel(Γ); furthermore, Bel(Γ)+Bel(W\Γ) = 1. Only in the caseswhere I > 0, Bel(Γ)+Bel(W\Γ)< 1 and P(Γ)+Bel(W\Γ)> 1.


This section is organized as follows: section 1.2.2.1 describes the structure of quanti-fied statements expressed as conditional statements using Dempster-Shafer’s Theory; sec-tion 1.2.2.2 describes the proposed syllogistic inference schemas and, finally, section 1.2.2.3summarizes the main points of this framework emphasizing its strengths and weakness.

1.2.2.1 Theory of Support Logic Categorical Statement

Categorical statements in Support Logic follow the standard form of quantified statementsexpressed as conditional statements, i.e., if x is A then x is B [a,b], where [a,b] denotes thestrength that links the antecedent with the consequent, if x is A the subject-term and then x

is B the predicate-term. In this case, the strength is the degree of belief of the speaker inthe truthfulness of the proposition and constitutes its degree of support. Precise quantifiersdenote precise values of belief and fuzzy quantifiers fuzzy ones. For instance, “most basketplayers are tall” is equivalent to “if a human being is a basket player then he is tall with asuggestive evidence that is most”. “Most” conveys a high quantity or a strong link betweenbasket players and tall people, therefore, given that it is a fuzzy quantifier, an interpretationis the interval most = [0.8,1]. Maximum degree of support is denoted by the quantifier “all”and minimum support is denoted by the quantifier “some not”.

Bel(Γ) and P(Γ) constitute the support interval of propositions and it is represented troughthe interval [α,b], where α stands for Bel(Γ) and b stands for P(Γ). Furthermore, it is relevantto note that Bel(Γ)≤ P(Γ). Thus, each one of the values of the support interval has associateda measure of probability with the following meanings:

Bel(Γ) It is the lower bound. It denotes a subjective probability; i.e., how is the relevant forthe speaker the evidence that supports a particular event or fact.

P(Γ) It is the upper bound. It denotes an objective probability; i.e., the plausibility of theevent in relation with the other contemporary facts or events.

Thus, considering again the example “if a human being is a basket player then he is tallwith a suggestive evidence that is most”, we can assign to the quantifier “most” the supportvalue [0.8,1], given a usual interpretation for this quantifier, obtaining “if a human being is abasket player then he is tall [0.8,1]”.


PR1 If q then r [a,b]PR2 If p then q [c,d]

C If p then r [ac,1− c(1−b)]

Table 1.44: Chaining/Property Inheritance based on Support Logic.

PR1 If p then q [a,b]PR2 If p∧q then r [c,d]

C If p then r∧q [b(1−a(1−d))]

Table 1.45: Intersection/Product based on Support Logic.

1.2.2.2 Theory of Support Logic Syllogism

In order to build a full system for syllogistic reasoning, Spies defines a negation operator anda two-place connective:

Negation operator This is defined to negate the support values of a statement. Let us con-sider the statement A[a,b]; Bel(Non−A) = 1−P(A) and P(Non−A) = 1−Bel(A); thecorresponding support interval is [1−b,1−a].

Two-place connective This is a combination for assessing the “joint” degrees of belief andplausibility of a set of statements that constitute the premises and the conclusion of acertain syllogism. Of course, all the statements involved in a syllogism must be definedby the corresponding support intervals.

Spies [79] deals with four of Zadeh’s patterns (see 1.1.5.2), including asymmetric andsymmetric ones. The notation used is explained as follows: p,q,r stands for propositions andthe particles if. . . then denotes the typical form a conditional statement:

– Chaining or Property Inheritance, it is an asymmetric syllogism. Table 1.44 shows itschema.

– Intersection/Product, it is also an asymmetric syllogism. Table 1.45 shows its struc-ture.

– Antecedent Conjunction, it is a symmetric syllogism. Two additional constraints mustbe assumed for reasoning with it: i) p[1,1] and q[1,1]; ii) v and w are computed accord-ing to Dempster’s rule applied to the granules with support pairs of [a,b] and [c,d],respectively. Table 1.46 shows its form.


PR1 If p then r [a,b]PR2 If q then r [c,d]

C If p∧q then r [v,w]

Table 1.46: Antecedent Conjunction based on Support Logic.

PR1 If r then p [a,b]PR2 If r then q [c,d]

C If r then p∧q [ac,bd]

Table 1.47: Consequent Conjunction based on Support Logic.

– Consequent Conjunction; this is a symmetric syllogism. Table 1.47 shows its schema.

1.2.2.3 Summary

Support Logic Syllogistics also adopts a conditional reading of quantified statements but mea-suring the strength between the antecedent and the consequent with Dempster-Shafer’s theoryof evidence. From an epistemological point of view, it is more intuitive than the probabilitytheory as it evaluates the subjective evidence that supports a statement but, from an intersub-jective point of view, its results are less intuitive as the individual motivations for a belief arenot always transparent.

Attending to inference schemas, it is not a solid model. Although it deals with four rea-soning patterns comprising by three terms, two premises and a conclusion, it is not compatiblewith Aristotelian Syllogistics as it only deals with Zadeh’s schemas which, as we showed insection 1.1.5, are incompatible with classical moods. Furthermore, only the use of propor-tional quantifiers is considered and, while the results of the Probabilistic Syllogistics matchwith the set-based approaches, Support Logic Syllogistics can give way to different results.

1.2.3 Qualified Syllogistics

D. Schwartz [71] analyses the syllogism within his general framework of Dynamic SystemReasoning (DRS), a global approach to the field of non monotonic reasoning. Schwartz de-fines it as [71, p. 103]:


Quantification Usuality Likelihoodall always certainlyalmost all almost always almost certainlymost usually likelymany/about half frequently/often uncertain/about 50-50few/some ocasionally/seldom unlikelyalmost no almost never/rarely almost certainly notno never certainly not

Table 1.48: Interrelations across several level of modifiers.

A qualified syllogism is a classical Aristotelian syllogism that has been “qual-ified” through the use of fuzzy quantifiers, likelihood modifiers, and usualitymodifiers.

It is worth noting that Schwartz’s framework opens syllogism up to likelihood and usualitymodifiers and not only consider quantifiers. Each one of them has different uses in naturallanguage, depending on what it is intended to be said:

– Quantifiers: They are used for describing current affairs or a present state of the world.For instance, “today, most birds can fly”.

– Usuality modifiers: They are used to show a regularity in past experiences. For instance,“usually birds fly”.

– Likelihood modifiers: They are used to make predictions about the future or for describ-ing our expectations about it. For instance, “if at some future time I randomly select abird, it is likely that it will be able to fly”.

The interrelationships across several levels of this type of modifier [71, p.118] are shownin Table 1.48.

These relationships also can be transferred to the field of reasoning, as “there is a naturalconnection between fuzzy quantification and fuzzy likelihood [. . . ]. The foregoing two con-cepts are also connected with the concept of usuality” [71, pp.116-117] through the notion ofstatistical sampling.

This section is organized as follows: section 1.2.3.1 describes the form of qualified cate-gorical statements; section 1.2.3.2 describes the main characteristics of his inference schemasand, finally, section 1.2.3.3 summarizes the strengths and weakness of this model.


PR1 Most men are vainPR2 Socrates is a manC It is likely that Socrates is vain

Table 1.49: Example of qualified syllogism.

1.2.3.1 Theory of Qualified Categorical Statements

The structure of Qualified Categorical Statements does not include relevant additional contri-butions with respect to the conditional reading proposals described in sections 1.2.1 and 1.2.2.However, he also distinguishes between the two semantics explained in this chapter; i.e., con-ditional interpretation and set-based one:

– Bayesian semantics: It is supported on the Bayesian framework of probability and itis the most general one. Therefore, the notion of conditional probability is assumed asprimitive and also a subjectivist theory of probability is adopted.

– Counting semantics: It implicitly refers to a domain of discourse. For instance, “mostbirds can fly” refers to a domain that includes a collection of birds. The subjectivist the-ory of probability and the notion of conditional probability are avoided and substitutedby the standard definition of probability: events are represented as subsets of a universeof alternative possibilities. In the fuzzy case, Σ− count is the most used proposal.

We classify this approach as conditional reading as Schwartz says that Bayesian semanticsis the most general one.

1.2.3.2 Theory of Qualified Syllogism

Schwartz explicitly talks about Aristotelian syllogisms, but his proposal is based on a type ofargument that is not properly a syllogism. Table 1.49 shows the typical example of qualifiedsyllogism defined by Schwartz.

This argument does not follow the minimum characteristics of Aristotelian syllogisms as itinvolves a singular statement (PR2, “Socrates is a man”) instead only categorical ones, sinceSocrates refers to an individual and not to a set. We can think that Schwartz assumes theequivalence between singular statements and A statements, but there is no reference to thisquestion or to the existential import. As a consequence, there are not three terms involved inthe syllogism but only two; i.e., men and vain. Finally, the inference pattern underlying this


PR1 If x is S, then x is P [m]PR2 a is SC a is P[m]

Table 1.50: Canonical form of qualified syllogism.

example is based on Modus Ponens (see Table 1.50), although the explained modifiers ([m])are incorporated.

1.2.3.3 Summary

Qualified Syllogistics is only a small part of a global approach to DRS. Its main contributionis that of establishing relationships of equivalence between quantifiers, usuality and likelihoodmodifiers. However, its analysis of quantifiers is reduced to a small set of labels as examples,without additional considerations about them. Furthermore, there is no clear difference be-tween categorical and singular statements, something fundamental in the previous models ofsyllogism.

From the point of view of reasoning pattern, he directly considers syllogism as a particularcase of Modus Ponens, which is modified introducing the modifiers previously defined.

1.3 Discussion: Set-Based Interpretation vs. Conditional Inter-pretation

Syllogistic reasoning can be briefly defined as reasoning about quantities (fuzzy or not) thatare usually expressed as quantified statements. Thus, this chapter has sought to present asummary of the main approaches to syllogistic reasoning developed in the history of logicand lay the foundations for a further development.

We have described two different frameworks for managing this kind of inferences: theso-called set-based interpretation and the conditional one. The difference between them is inthe analysis of quantified statements, although both schemes start from the same basic patternQ S are P, where Q is the quantifier, S the subject and P the predicate.

Thus, the set-based approach establishes that S and P are sets of elements (known as term-sets) and the quantifier Q describes the relationship between both sets in terms of quantities(fuzzy or not); for instance, “most basket players are tall” means that a number of elements bigenough of the set basket players belongs to the set of tall people. Since this is a relationship

1.3. Discussion: Set-Based I. vs. Conditional I. 97

between sets, we are talking about a second order logic. Following this idea, we have pre-sented six approaches which are the most relevant ones from our point of view. Aristotelian,Intermediate and Exception Syllogistics explores syllogistic reasoning considering crisp andvague quantifiers but out of a fuzzy framework while Interval, Fuzzy and Generalized Inter-mediate Syllogistics manage vague quantifiers using fuzzy logic.

Aristotelian Syllogistics is the first system in the history of logic and laid the foundationsfor the subsequent developments. He defines categorical statements as the only valid propo-sitions for syllogistic reasoning, where the S and P are term-sets and Q the quantifier. He alsodefines the four quantifiers valid for making inferences (“all” and “no” are the universal onesand “some” and “some. . . not” are the particular or existential ones) and how they are linkedbetween themselves (the LSO). The basic structure of the inference pattern is the chaining oftwo terms (the so called NT and MT) by a middle one (the so called DT); therefore, everysyllogism comprises three terms, two premises and a conclusion. The combination of theposition of the middle term in the premises (the so-called Figures) with the quantifiers of theLSO generates the 24 Aristotelian moods.

In twentieth century, coinciding with a recovery of Aristotelian thinking, an expansion ofAristotelian Syllogistics was proposed by introducing the concept of intermediate quantifier.This is defined as any quantifier between the universal and the particular ones such as “most”,“few”, “many”, etc. Intermediate Syllogistics produces new LSOs with more quantifiers and,hence, new syllogistic moods appear. Furthermore, it also opens up the possibility of an alge-braic method supported by Venn Diagrams to calculate the quantifier of the conclusion, thuscomplementing the classical proof method based on logical laws. This idea is consolidatedin Exception Syllogistics, where a full algebraic method for syllogism is proposed. In [77]another algebraic method for syllogism appears, but only for the four Aristotelian quantifiers,and its results can be integrated in Exception Syllogistics. In addition to this method, Ex-ception Syllogistics also introduces exception quantifiers (such as “all but three”, “at mostfive”, etc.), a relevant innovation with respect to the Intermediate Syllogistics (only focusedon proportional quantifiers), and the combination thereof with proportional ones.

In the fuzzy field, Interval Syllogistics introduce the representation of quantifiers as inter-vals, an intermediate step to a full fuzzy one. In this case, linguistic quantifiers are numericallyinterpreted and the necessity of an LSO is avoided. On one hand, is a more precise proposal asit can manage precise and imprecise quantifiers simultaneously but on the other hand, it losesa degree of interpretability as the translation from natural language to percentages or intervals


is not unambiguous. In this system, the quantifier conclusion is obtained through a calculationprocedure where the premises are the constraints and the conclusion must be compatible withthem. In this way, the concept of logic proof is not given up. Fuzzy Syllogistics follows thisvein exactly, but using the following fuzzy definition for the quantifiers: each one has a fuzzynumber associated to it and the quantifier of the conclusion is directly calculated using fuzzyarithmetic. Finally, Generalized Intermediate Syllogistics is a generalization of IntermediateSyllogistics to a fuzzy framework, but without relevant innovations in either in the quantifiersmanaged (although there is it in how they are defined) or in the syllogistic inferences schema.

In general, set-based approaches stand out in dealing with quantified statements as asser-tions; i.e., propositions that affirm or deny something although they involve fuzzy quantifiers.This is a relevant question from a pragmatic point of view, since an assertion is different ofa rule or a condition. Furthermore, all these frameworks assimilate the logical form of thestatements to their syntactical form, making their analysis easier. Non-fuzzy frameworks alsoinvolve relationships of opposition typical in natural language as logical rules (the links inthe LSO), while fuzzy frameworks substitute them with the concepts, such as order or nega-tion, which are also very typical in natural language. Another interesting innovation in fuzzymodels is the transition from the syllogism based on logic proofs to a heuristic syllogismbuilt using an algebraic method for calculating the quantifier of the conclusion in terms of thequantifiers of the premises.

Nevertheless, set-based approaches also have a number of weaknesses that must be con-sidered. Firstly, the question of existential import of universal quantifiers, which has beenunder debate since the Middle Ages. There is not one single solution, although the mostwidely accepted one is to assume it as an additional restriction of the syllogism, saying thatthe subject-term of universal propositions is not empty. Secondly, introducing statements in-volving singular terms such as “Socrates” or “S. Holmes” into a syllogism is a problematicquestion, although from an intuitive point of view, inferences such as “all human beings aremortal, Socrates is a human being; therefore, Socrates is mortal” are very clear. The acceptedsolution is to transform singular statements into categorical statements qualifying the singu-lar term with the universal quantifier “all” to emphasize that it is unique and transformingit into a categorical term; i.e., “all human beings are mortal, all Socrates is a human being;therefore, all Socrates is mortal”. However, from the point of view of natural language use,the statement is strange. Finally, other weak point of fuzzy approaches (except Generalized

1.3. Discussion: Set-Based I. vs. Conditional I. 99

Intermediate Syllogistics) is that their compatibility with the Aristotelian one is only partial,although Interval Syllogitiscs shows correct behaviour.

Conditional interpretation assumes that quantified statements can be rewritten as impli-cations, where S is the antecedent, P the consequent and Q denotes the strength of the linkbetween S and P. For instance, “most basket players are tall” is equivalent to “if somebody is

a basket player then somebody is tall with most support”. The main difference with respectto set-based interpretation is that properties and individuals are distinguished here; i.e., interms of quantified statements, there is no difference between categorical and singular state-ments. This supplements one of the weak points of the other syllogistic frameworks. Thethree frameworks analysed following this interpretation differ in their semantics to quantifiedstatements: Probabilistic Syllogistics assumes that a quantifier expresses probability; SupportLogic Syllogistics assumes that it express belief; and Qualified Syllogistics assumes that itcan express quantity, temporality or probability.

Probabilistic Syllogistics is based on the psychology of reasoning rather than on logic.This leads it to analyse syllogistic reasoning as a heuristic procedure in terms of Bayesianreasoning instead of as a logic one, substituting the concept of logic validity for reliability.It makes a deep analysis of Aristotelian Syllogistics and the compatibility between them isproved. Examples of intermediate quantifiers are also considered and the different psycho-logical strategies for making inferences are explained. Support Logic Syllogistics substitutesprobability theory for Dempster-Shafer’s theory of evidence. Aristotelian Syllogistics is notcompatible with this system as only Zadeh’s inference schemas (Fuzzy Syllogistics) are con-sidered. Finally, Qualified Syllogistics introduces the suggestive idea about the links or equiv-alences between quantifiers, usuality and likelihood modifiers, although syllogistic argumentsare transformed into Modus Ponens substituting the Minor Premise of the syllogism for asingular statement.

In general, the approaches based on conditional interpretation deal with the combinationof categorical statements and singular statements better than set-based approaches and facil-itate the analysis of this kind of arguments, although they are not strictly syllogistic. Theproblematic question of existential import is also avoided as the statements managed are notassertions but rules in conditional form. Furthermore, Probabilistic Syllogistics uses Bayesianprobability, which is well known and its results are very successful. Nevertheless, all theseapproaches only deal with proportional quantifiers; other quantifiers, such us absolute, excep-tion or comparative, are explicitly rejected, as we have explained in Probabilistic Syllogistics.


On the other hand, the transformation of quantified statements into conditional propositionsalso entails transforming assertions into rules or conditions generating certain imprecision orvagueness not present in the original statement. For instance, “all students are tall” describesthe fact the all students are tall19 while “if somebody if a student then is sure that is tall”describes a hypothetical fact that is test when somebody in particular is chosen. Finally, therelationships of the LSO cannot be adequately explained using a probabilistic interpretationeither, as the different kind of oppositions are not considered.

For all the reasons explained, the framework defined by set-based interpretation seemsto be the most suitable for the purposes of our research: developing a proposal of syllogis-tic reasoning that is compatible with TGQ and flexible for dealing with different patterns ofinference that can appear in natural language. We mainly focus on the development of a strat-egy for calculating the quantifier of the conclusion in terms of the quantifiers of the premisesrather than a logic test to check its validity. In chapter 2 we go on to describe the model thatwe propose to achieve this aim.

19We are assume the Tarki’s Theory of Truth [80].

CHAPTER 2

SYLLOGISTIC REASONING USING THE

THEORY OF GENERALIZED QUANTIFIERS

In the previous chapter, different frameworks for dealing with syllogistic reasoning were anal-ysed. In this analysis, it is possible to see the evolution from the twenty-four Aristotelianmoods, managing only four quantifiers and four inference patterns, to fuzzy approaches,which can manage vague quantifiers and more than a hundred inference patterns, where thechaining is not only based on a middle term but also in other logic operations, such us con-junction and disjunction. However, we have also concluded that there are a number of weakpoints that can be improved to increase natural language arguments that can be considered:i) other quantifiers that we manage into natural language must be incorporated into syllogis-tic reasoning; ii) the combination of different kind of quantifiers in the same syllogism; iii)dealing with complex arguments involving m terms, n premises and p conclusions; and, iv)managing crisp and fuzzy definitions of different kind of quantifiers.

As constraints, we assume that Aristotelian Syllogistics must be included as a particularcase of this broad syllogistic framework and the compatibility with TGQ, the current linguistictheory for dealing with quantifiers in natural language. Furthermore, both challenges are notincompatible given the relationship between them explained by D. Westersthål [86].

As we have already stated, we consider the set-based interpretation to be the one bestsuited to our aims. The main reason for this choice is its compatibility with the TGQ [8]. Itis the standard theory accepted by logicians and linguistics for dealing with quantifier expres-sions and analysing quantification in natural language combining the logic and the linguistics

102 Chapter 2. Syllogistic Reasoning Using the TGQ

perspectives. Its key concept is the generalized quantifier, understood as a second order pred-icate that establishes a relationship between two classical sets. Assuming this theory opensup the possibility of going much further than the usual absolute/relative quantifiers. With re-gard to the reasoning process, a heuristic perspective is adopted rather than a logical one as itmakes it easier to deal with arguments involving more than three terms and two premises.

This chapter is organized as follows: section 2.1 describes the structure of the statementsthat are used in our proposal of syllogistic framework; section 2.2 describes our proposal ofreasoning schema and inference mechanism; section 2.3 shows some illustrative examplesabout how our framework works; section 2.4 explains the combination between absolute andproportional quantifiers; and, finally, section 2.5, summarizes the main characteristics of thisframework and analyses its strengths and weakness.

2.1 Theory of Categorical Statements Using the Theory of Gen-eralized Quantifiers

As we have already stated, the standard form of the quantified statements valid for syllogismaccording to set-based interpretation is Q S are P, where Q is the quantifier, S the subject-termand P the predicate-term. Depending on the particular syllogistics, Q could adopt differentforms, although always binary (taking two sets as arguments); for instance, in IntermediateSyllogistics it is quantifiers such as “most”, “few”, etc. while in Exception Syllogistics itis terms such as “all but four”, “at least two”, etc. Our contribution in this topic is to useTGQ to define the form of Q. As examples, we describe the case of proportional, absolute andexception quantifiers in detail.

Considering the proportional interpretation of the quantifier “all”, and denoting E as thereferential universe and P(E) as the power set of E, the evaluation of the quantified statement“All Y1 are Y2” for Y1,Y2 ∈P(E) can be modelled as:

all : P(E)×P(E)→{0,1}

(Y1,Y2)→ All(Y1,Y2) =

{0 : i f Y1 6⊆ Y2

1 : i f Y1 ⊆ Y2

2.1. Theory of Categorical Statements Using the TGQ 103

For absolute quantifiers, evaluation of a statement like “Between 3 and 6 Y1 are Y2” (Bet3,6)can be modelled as:

Bet3,6 : P(E)×P(E)→{0,1}

(Y1,Y2)→ Bet3,6(Y1,Y2) =

{0 : i f |Y1∩Y2| /∈ [3,6]

1 : i f |Y1∩Y2| ∈ [3,6]

For quantifiers of exception, let us consider the sentence “All but three Y1 are Y2”; forY1,Y2 ∈P(E). Its evaluation can be modelled as:

All but three : P(E)×P(E)→{0,1}

(Y1,Y2)→ All but three(Y1,Y2) =

{0 : i f |Y1∩Y2| 6= 3

1 : i f |Y1∩Y2|= 3

In general, for any quantified proposition whose quantifier is a binary one from TGQ; letP= {Ps, s = 1 . . . ,S} be the set of relevant properties defined in E. For instance, P1 can denotethe property “to be a student” and P2 “to be tall”. Let L1 and L2 be any Boolean combinationof the properties in P. Thus, the typical form of our categorical statements is:

Q L1 are L2 (2.1)

where Q denotes a binary quantifier of TGQ, L1 is the subject-term or restriction and L2 thepredicate-term or scope. It is worth noting that L1 and L2 can be any Boolean combinationof P. Thus, we can incorporate to our proposal the inferences based on logic operators (theso-called symmetric syllogisms). We consider the following types of quantifiers:

– Logical quantifiers (QLQ): The four Aristotelian ones, “all”, “no”, “some”, “not all”(“some. . . not”).

– Absolute Binary quantifiers (QAB): Natural numbers with or without some type ofmodifier such us “around five”, “more than twenty-five”, etc.) following the generalpattern QAB Y1 are Y2 (e.g., “around five students are tall”).

– Proportional Binary quantifiers (QPB): Linguistic terms such us “most”, “almost all”,“few”, etc. following the general pattern QPB Y1 are Y2 (e.g. “few students are tall”).


– Binary Quantifiers of Exception (QEB): Proportional binary quantifiers with a naturalnumber such us “all but three”, “all but three or four”, etc. following the general patternQEB Y1 are Y2 (e.g., “all but five students are tall”).

– Absolute Comparative Binary Quantifiers (QCB−ABS): Natural numbers with modi-fiers of type more, less, such us (“three more. . . than”, “four less. . . than”, etc.) follow-ing the general pattern There is/are QCB−ABS Y1 than Y2 (e.g. “there are three more boysthan girls”).

– Proportional Comparative Binary Quantifiers (QCB−PROP): Rational multiple or par-titive numbers, such us “double”, “half”, etc. following the general pattern There is/are

QCB−PROP Y1 than Y2 (e.g., “there are double boys than girls”).

– Similarity Quantifiers (QS): Linguistic expressions that denote similarity (S) betweentwo given sets “very similar”, “few similar”, etc. following the general pattern A and

A′ are very/few/. . . similar; where A denotes one of the sets of the comparison and A′

denotes a set similar to A (e.g. “the audience of opera and ballet are very similar”).

Table 2.1 summarizes the definition of these quantifiers according to the TGQ.

2.2 Theory of Syllogism Using the Theory of Generalized Quan-tifiers

In the analysis conducted in chapter 1, we pointed out that some models, such as AristotelianSyllogistics, focus on developing strategies to prove the logical validity of the syllogisms,while other models, such us Exception Syllogistics or Interval Syllogistics, focus on devel-oping methods to calculate the conclusion in terms of the quantifiers of the premises. Thefact that the former only consider one conclusion by argument and the latter consider sev-eral different possibilities justifies these different strategies. Furthermore, as pointed out byProbabilistic Syllogistics, this second strategy, which we qualify as heuristic, is also typicalin human reasoning.

Thus, assuming a heuristic challenge and adhering mainly to Interval Syllogistics (seesection 1.1.2), as it has shown the best behaviour for dealing with Aristotelian Syllogistics, wepropose a transformation of the syllogistic reasoning into an equivalent optimization problemthat consists of calculating the extreme quantifiers of the conclusion taking as restrictions thequantifiers of the premises.

2.2. Theory of Syllogism Using the TGQ 105

Logical quantifiers

QLQ−all(Y1Y2) =

{0 : Y1 6⊆ Y21 : Y1 ⊆ Y2

QLQ−no(Y1,Y2) =

{0 : Y1∩Y2 6= /01 : Y1∩Y2 = /0

QLQ−some(Y1,Y2) =

{0 : Y1∩Y2 = /01 : Y1∩Y2 6= /0

QLQ−not−all(Y1,Y2) =

{0 : Y1 ⊆ Y21 : Y1 6⊆ Y2

Absolute binary quantifiers

QAB(Y1,Y2) =

{0 : |Y1∩Y2| 6= N1 : |Y1∩Y2|= N

Proportional binary quantifiers

QPB(Y1,Y2) =

0 : |Y1∩Y2|

|Y1|/∈ LT

1 : |Y1∩Y2||Y1|

∈ LT1 : |Y1|= 0

Exception binary quantifiers

QEB(Y1,Y2) =

{0 : |Y1∩Y2| /∈ N∗1 : |Y1∩Y2| ∈ N∗

Comparative binary quantifiers

QCB−ABS(Y1,Y2) =

{0 : |Y1|− |Y2| /∈ N∗1 : |Y1|− |Y2| ∈ N

QCB−PROP(Y1,Y2) =

{0 : |Y1|

|Y2|/∈ Q∗

1 : |Y1||Y2|∈ Q∗

Similarity quantifiers

QS(Y1,Y2) =

0 : |Y1∩Y2|

|Y1∪Y2|< S,Y1∪Y2 6= /0,S /∈ Q∗

1 : |Y1∩Y2||Y1∪Y2|

≥ S,Y1∪Y2 6= /0,S ∈ Q∗1 : Y1∪Y2 = /0

Table 2.1: Definition of some binary quantifiers according to TGQ.


PR1: Q1 L1,1 are L1,2PR2: Q2 L2,1 are L2,2. . .

PRN: QN LN,1 are LN,2C: QC LC,1 are LC,2

Table 2.2: Inference scheme of broad syllogism.

(a) AAA mood.

PR1: Qall L1,1 are L1,2PR2: Qall L2,1 are L1,1

C: Qall L2,1 are L1,2

(b) AII mood.

PR1: Qall L1,1 are L1,2PR2: Qsome L1,1 are L2,1

C: Qsome L2,1 are L1,2

Table 2.3: Aristotelian moods expressed using broad inference pattern.

2.2.1 Inference Pattern

To achieve our challenge of developing an inference pattern flexible enough to deal withcomplex arguments composed of more than three terms and two premises, we have to dealwith all the inference patterns developed by the different proposals of syllogisms.

So, let us now consider the form defined in the previous section to categorical statements,our proposal of broad inference pattern for N premises, PRn,n = 1, . . . ,N and a conclusion,C, is described in Table 2.2; where Qn, n = 1,. . . ,N are the linguistic quantifiers in the N

premises; Ln, j, n = 1, . . . ,N, j = 1, 2 denotes an arbitrary Boolean combination between theproperties considered in the syllogism, QC stands for the quantifier of the conclusion and LC,1

and LC,2 stand for the subject-term and the predicate-term in the conclusion.Let us mention a number of patterns in the analysed frameworks to syllogistic reasoning

to show how this schema can be adapted.From Aristotelian Syllogistics we take the AAA mood from Figure I and the AII mood

from Figure III, the so-called axioms of this system according to J. Łukasiewicz [33]. Thus,since this framework only manages three terms, we have P3; two premises, PR2; a conclusionC and the four logic quantifiers, Qn ∈ {all, none, some, some. . . not}. Therefore, Ln, j is withn = 2 and j = 2. Table 2.3 shows both schemas.

From Intermediate Syllogistics, we take, for instance, ETD from Figure II. As in Aris-totelian Syllogistics, P3, PR2, C and Qn ∈{all, almost all, most, many, no, few, most. . . not,many. . . not, some. . . not}. Therefore, Ln, j is with n= 2 and j = 2. Table 2.4 shows its schema.


PR1: Qnone L1,1 are L1,2PR2: Qmost L2,1 are L1,2

C: Qmost L2,1 are not(L1,2)

Table 2.4: ETD (Figure II) from Intermediate Syllogistics expressed using broad inferencepattern.

PR1: Qat−least−all−but−0 L1,1 are L1,2PR2: Qat−least−1 L1,1 are L2,1

C: Qat−least−1 L2,1 are L1,2

Table 2.5: AI- (Figure III) from Exception Syllogistics expressed using broad inference pat-tern.

PR1: Q1 L1,1 are L1,2PR2: Q2 L1,2 are L1,1PR3: Q3 L1,2 are L3,2PR4: Q2 L1,1 are L3,2

C: QC (L1,1 and L1,2) are L3,2

Table 2.6: Pattern II from Interval Syllogistics expressed using broad inference pattern.

From Exception Syllogistics, we choose, for instance, the AI- schema of Figure III, thatis, the corresponding to the AII schema of Aristotelian Syllogistics. In this case, P and PR

are the same, but C can be multiple. The quantifiers are Qn ∈ {at least all but x, at most x, atleast x, at least x. . . not} where x depends on the context and the cardinality of Ln,1. Table 2.5shows the corresponding pattern, where C is the minimal conclusion.

All of these patterns are of asymmetric type; i.e., based on a chaining through a middleterm. They only use negation as Boolean operation on the predicate of the statements. Fuzzyframeworks are the first ones to introduce symmetric syllogisms; i.e., those whose chainingis through logic operations such as conjunction or disjunction. This means that subject-termsand predicate-terms involve more Boolean operations than asymmetric frameworks. We shallnow show some examples of how these inference patterns can be implemented in our proposalof broad reasoning pattern.

Hence, from Interval Syllogistics we took Pattern II, which involves a conjunction in thesubject of the conclusion. Thus, P3, PR4, C and Qn = [q,q] with [q,q]∈ [0,1]. Table 2.6 showsthe corresponding schema.


PR1: Q1 L1,1 are L1,2PR2: Q2 (L1,1 and L1,2) are L2,1

C: QC L1,1 are ((L1,2)and L2,1)

Table 2.7: Intersection/product from Fuzzy Syllogistics expressed using broad inference pat-tern.

From Fuzzy Syllogistics, we took the most basic pattern defined by Zadeh, Intersec-tion/product syllogism. In this case, P3, PR2, C and Qn are fuzzy numbers. The Booleanoperator is a conjunction in the subject of PR2. Table 2.7 shows the corresponding schema.

In order to explain the inference mechanism to calculate C in terms of PR1 to PRN, wedistinguish three steps: i) dividing the universe of discourse into disjoint sets; ii) definingsentences as systems of inequations; iii) selecting the optimization method that should beapplied in each case in order to resolve the reasoning process.

2.2.2 Division of the Universe in Disjoint Sets

Following the ideas explained in sections 1.1.2 and 1.1.3, the referential universe E is parti-tioned into a new set of disjoint sets PD = {P′1, . . . ,P′K} with K = 2S containing the elementsin E that fulfil or not the S terms or properties defined in a syllogism in the following way:

P′1 = P1∩P2∩ . . .PS−1∩PS

P′2 = P1∩P2∩ . . .PS−1∩PS

. . .

P′K = P1∩P2∩ . . .PS−1∩PS

(2.2)

where Ps = {e ∈ E, e f ul f ills Ps} and Ps = {e ∈ E, e does not f ul f ill Ps}. Therefore,K⋂

k=1

P′k =

/0 andK⋃

k=1

P′k = E.

Figure 2.1 shows an example for S = 3 properties, where the P′k, k = 1, . . . ,8 denote eachone of the disjoint sets that are generated from the properties P1,P2 y P3. Thus, for instanceP′8 = P1∩P2∩P3, P′2∪P′6 = P2∩P3 or P1=P′5∪P′6∪P′7∪P′8.


P'5P'7

P'3

P'6

P'8P'4

P'2

P'1

E

P3

P1 P2

Figure 2.1: Division of the referential universe for the case of S = 3 properties and 8 = 23

elements in the partition P′k, k = 1, . . . ,8. P1 = P′5 ∪P′6 ∪P′7 ∪P′8;P2 = P′3 ∪P′4 ∪P′7 ∪P′8;P3 =P′2∪P′4∪P′6∪P′8.

2.2.3 Transformation into Inequations

The syllogistic reasoning problem is transformed into an equivalent optimization problem,disregarding the type of quantifiers involved; notwithstanding, we can deal at least with alltypes of quantifiers compatible with the linguistic TGQ (defined in Table 2.1).

We describe the transformation with quantifiers defined as crisp intervals (Q = [a,b]),since this is the basis for dealing with fuzzy quantifiers. For the sake of simplicity, we assumethat the Li, j are atomic; i.e., they are not Boolean combinations of sets. This assumptiondoes not limit the the generality of the solution that can be extended by simply and directlysubstituting the atomic parts by the Boolean combination of sets. For instance, for S = 2 wehave the following combinations of disjoint sets:

P′1 = L1∩L2

P′2 = L1∩L2

P′3 = L1∩L2

P′4 = L1∩L2

(2.3)

Where the usual notation L1,L2 for the terms in the propositions has been used instead ofthe one introduced in expression 2.2.


Logical definition Equivalent inequation

QLQ−all(Y1Y2) =

{0 : Y1 6⊆ Y21 : Y1 ⊆ Y2

x2 = 0

QLQ−no(Y1,Y2) =

{0 : Y1∩Y2 6= /01 : Y1∩Y2 = /0 x3 = 0

QLQ−some(Y1,Y2) =

{0 : Y1∩Y2 = /01 : Y1∩Y2 6= /0 x3 > 0

QLQ−not−all(Y1,Y2) =

{0 : Y1 ⊆ Y21 : Y1 6⊆ Y2

x2 > 0

QAB(Y1,Y2) =

{0 : |Y1∩Y2| /∈ [a,b]1 : |Y1∩Y2| ∈ [a,b] x3 ≥ a;x3 ≤ b

QPB(Y1,Y2) =

0 : |Y1∩Y2|

|Y1|< a∨ |Y1∩Y2|

|Y1|> b

1 : |Y1|= 01 : |Y1∩Y2|

|Y1|≥ a∧ |Y1∩Y2|

|Y1|≤ b

x3x2+x3

≥ a;x3

x2+x3≤ b

QEB(Y1,Y2) =

{0 : |Y1∩Y2| /∈ [a,b]1 : |Y1∩Y2| ∈ [a,b]

x2 ≥ a;x2 ≤ b

QCB−ABS(Y1,Y2) =

{0 : |Y1|− |Y2| /∈ [a,b]1 : |Y1|− |Y2| ∈ [a,b]

(x3 + x2)− (x3 + x4)≥ a;(x3 + x2)− (x3 + x4)≤ b

QCB−PROP(Y1,Y2) =

{0 : |Y1|

|Y2|/∈ [a,b]

1 : |Y1||Y2|∈ [a,b]

(x3+x2)(x3+x4)

≥ a; (x3+x2)(x3+x4)

≤ b

QS(Y1,Y2) =

0 : |Y1∩Y2|

|Y1∪Y2|< a,Y1∪Y2 6= /0

1 : |Y1∩Y2||Y1∪Y2|

≥ a,Y1∪Y2 6= /01 : Y1∪Y2 = /0

x3x2+x3+x4

≥ a; x3x2+x3+x4

≤ b

Table 2.8: Definitions of interval crisp Generalized Quantifiers Q = [a,b]

Denoting xk = |P′k|, k = 1, . . . ,K we therefore have,

x1 =∣∣L1∩L2

∣∣x2 =

∣∣L1∩L2∣∣

x3 = |L1∩L2|

x4 =∣∣L1∩L2

∣∣(2.4)

Now, the evaluation of any quantified proposition is equivalent to solving an inequationwhere the previously defined cardinality values xk,k = 1, . . . ,K are involved as variables. Ta-ble 2.8 (right) summarizes the inequations corresponding to each quantifier.

Let us consider the following example, “between three and six Y1 are Y2”. It entails the ab-solute binary quantifier defined as an interval; that is QAB = [3,6]. According to the definitionsshown in Table 2.8, this means that x3 ≥ 3 and x3 ≤ 6 (see (2.3) and (2.4)).


2.2.4 Definition and Resolution of the Equivalent Optimization Problem

Once the inequations corresponding to the statements involved in a syllogism have been de-fined, we are in a position to approach the reasoning problem as an equivalent mathematicaloptimization problem. The fundamental idea of this transformation is described in [17, 18]for binary proportional quantifiers and consists of defining the set of inequations associated tothe premises in the syllogism and solving it by applying of the appropriate resolution method(Simplex, fractional programming, etc.)[9]. However, our approach is more general since wecan also deal with all types of quantifiers and also any Boolean combination in the restrictionand scope of the quantified statements (premises and conclusions).

In order to correctly apply the resolution method, we need to add three additional con-straints to the set of inequations obtained from the syllogistic argument. The first one guar-antees that there are no sets with a negative number of elements; i.e., the existential import isassumed as a previous condition;

xk ≥ 0,∀k = 1, . . . ,K = 2S (2.5)

The other two constraints are only necessary if the quantifier of the conclusion is a pro-portional one. In this case, since the function to be optimized is a rational one, it should beguaranteed that;

– there are no 0 in the denominator of the involved fractions in order to avoid any indef-inition in the results; i.e., Ln,1 6= /0; thus if we denote by P′n,11 · · ·P′n,1r the disjoint partsof Ln,1, and by xn,1

r the cardinalities of the disjoint sets, the following must hold:

xn,11 + · · ·+ xn,1

r > 0,∀n = 1, ...,N (2.6)

– the sum of the cardinalities must equal the cardinality of the referential universe:K

∑k=1

xk = |E| (2.7)

The conclusion of the syllogistic argument is the statement QC LC,1 are LC,2 (as indicatedin 2.2). For the case of quantifiers of the absolute type (“no”, “some”, “some. . . not”, absolute,exception and comparative absolute), QC = [a,b] the expressions that have to be optimized arethe following ones:

a = minimize xm,n1 + . . .+ xm,nI (2.8)

b = maximize xm,n1 + . . .+ xm,nI (2.9)


with xm,ni ∈{xk,k = 1, ...,K}∀i= 1, ..., I subject to the following restrictions: the premisesof the syllogisms (from PR1 to PRN as in explained in section 2.2.3) and the restriction statedin (2.5). This kind of syllogism can be solved using SIMPLEX directly.

For the case of proportional quantifiers in the conclusion, techniques from fractional pro-gramming must be used. These are the quantifiers “all”, proportional, comparative propor-tional and similarity. The expressions to be optimized are:

a = minimizexm,n1 + . . .+ xm,nI

xm,d1 + . . .+ xm,dJ

(2.10)

b = maximizexm,d1 + . . .+ xm,nI

xm,d1 + . . .+ xm,dJ

(2.11)

with xm,ni ,xm,d j ∈ {xk,k = 1, ...,K}∀i = 1, ..., I,∀i = j, ...,J, and subject to the followingrestrictions: premises of the syllogism and the three additional constraints (2.5), (2.6) and(2.7).

Another contribution of our proposal to syllogistic reasoning is to deal with the combi-nation of absolute type quantifiers with proportional type ones in the premises of the sameargument. As we have stated, this idea was pointed to by N. Pfeifer [64], but only combiningexception and proportional quantifiers in the same statement and with the same form for allthe premises. We propose to go a step further; each premise can have its particular form anddifferent quantifiers in different propositions.

All Aristotelian moods except AAA of Figure I are good examples of this form of syllo-gism. According to our formalization of quantifiers, “all” has a proportional interpretation aswe assume that it refers to 100% meanwhile “no”, “some” and “some. . . not” have an abso-lute reading; i.e., “zero elements”, “at least one element” and “at least one element. . . not”.For instance, AII of Figure III involves “all” in the Major Premise and “some” in the MinorPremise and in the conclusion.

On the other hand, at pointed out in Exception Syllogistics (see section 1.1.3), the reason-ing with absolute quantifiers entails certain information on the cardinality of the subject-termof the premises. Given that our framework is flexible enough to introduce more than twopremises, this information can be introduced in a syllogism through additional premises withthe form Q S are S, where the subject-term and the predicate-term are the same and Q = QAB.

From the point of view of resolution procedure, certain problems with constraint (2.7) canappear as, in proportional terms, it is normalized to 1 or to 100% and an absolute quantifierhigher than them could generate a problem in the inequations. We propose using a very highvalue (i.e., 108, 109, 1010,. . . ) as the normalization value to avoid this problem.

2.3. Some Examples of Syllogisms Using the TGQ 113

The optimization procedure depends on the types of quantifiers that are being used. Inthe following sections we illustrate the behaviour of this proposal with different examples ofsyllogisms.

2.3 Some Examples of Syllogisms Using the Theory of General-ized Quantifiers

We stated that this framework was based on an interval interpretation of quantifiers, but it caneffectively deal with three different interpretations of the them:

– Crisp interval quantifiers: a crisp quantifier Q defined as an interval [a,b]. The case ofprecise quantifiers also can be managed as the particular case when a = b. For instance,“fifteen students are tall” the absolute quantifier “fifteen” is defined as QAB = [15,15].

– Fuzzy quantifiers approximated as pairs of intervals: a fuzzy quantifier Q defined asa pair of intervals {KERQ,SUPQ}, where KERQ = [b,c] represents the kernel andSUPQ = [a,d] the support of Q. For instance, “around fifteen students are tall”, theabsolute fuzzy quantifier “around fifteen” is defined as QAB = {[14,16], [13,17]}.

– Fuzzy quantifiers Q represented in the usual trapezoidal form with parameters [a,b,c,d].For instance, “around fifteen students are tall”, “around fifteen” is defined as QAB =

[13,14,16,17].

In the following sections, we describe the behaviour of our approach by using some ex-amples of fuzzy syllogisms.

2.3.1 Syllogisms with Crisp Interval Quantifiers

This is the simplest approach and the basis for solving the other two. The particular optimiza-tion technique to be applied depends on the type of quantifier in the conclusion. Absolute,exception and absolute comparative cases can be calculated using the SIMPLEX optimizationmethod, since the interval to be optimized depends on linear operations. In the case of pro-portional, comparative proportional and similarity quantifiers, linear fractional-programmingtechniques must be used [9].


PR1 : All animals but two are dogsPR2 : All animals but two are catsPR3 : All animals but two are parrots

C : There are QC animals

Table 2.9: Syllogism with quantifiers of exception.

2.3.1.1 Example 1: Dogs, Cats and Parrots

As a first example, we consider the following problem1:

Dogs, cats and parrots. How many animals do I have in my home, if all but twoare dogs, all but two are cats and all but two are parrots?

As we can see, a combination of exception crisp quantifiers are involved. In order to castthe wording into the typical form of a syllogism, the first step is to identify the number ofterms involved. In this case, for the sake of simplicity and coherence with Figure 2.1, let usconsider E = animals in my home and three terms dogs = P1, cats = P2 and parrots = P3.

Table 2.9 shows an initial formalization of the problem considering only the explicit in-formation of the wording. The corresponding quantifiers in PR1, PR2 and PR3 are Binary ofException (QEB) as defined in Table 2.8 (left) with [a,b] = [2,2].

According to Table 2.8 (right), each one of the premises generates the following systemof inequations:

PR1 :x1 + x2 + x3 + x4 = 2;

PR2 :x1 + x2 + x5 + x6 = 2;

PR3 :x1 + x3 + x5 + x7 = 2;

C :x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 = a;

(2.12)

where QC = [a,b] is the quantifier in the conclusion. Applying the SIMPLEX method, weobtain, nevertheless, QC = [0, in f ); i.e., an undefined result. The origin of this result is thatthe set of premises does not contain all the necessary information that human beings manageto solve this problem. It is necessary to incorporate additional premises with the implicit orcontextual information. In this case, we have added five premises (PR4−PR8) thus producingthe extended syllogism shown in Table 2.10.

1Problem taken from http://platea.pntic.mec.es/jescuder/mentales.htm


PR1 : All animals but two are dogsPR2 : All animals but two are catsPR3 : All animals but two are parrotsPR4 : No dog, cat or parrot is not an animalPR5 : No animal is not a dog, a cat or a parrotPR6 : No dog is a cat or a parrotPR7 : No cat is a dog or a parrotPR8 : No parrot is a dog or a cat

C : There are QC animals

Table 2.10: Syllogism with quantifiers of exception, logic and additional premises.

The corresponding quantifiers of the additional premises PR4−PR8 are logical quantifiersQ4 = Q5 = Q6 = Q7 = Q8 = no, labelled as QLQ−no in Table 2.8 (left). According to Table 2.8(right), the corresponding system of inequations is:

PR4 :x1 = 0;

PR5 :x1 = 0;

PR6 :x6 + x7 + x8 = 0;

PR7 :x4 + x7 + x8 = 0;

PR8 :x4 + x6 + x8 = 0;

C :x2 + x3 + x4 + x5 + x6 + x7 + x8 = a;

(2.13)

Applying the Simplex method, we obtain QC = [3,3]; that is, “There are three animals”;that is, I have a dog, a cat and a parrot in my home.

On the other hand, it is relevant to note that without the premises PR4 and PR5, we obtainQC = [2,3]; i.e., without these constraints we are assuming the possibility of the existence ofanimals in my home that are not dogs, cats or parrots. Nevertheless, in the wording of theproblem it is implicit that the only animals that can be at home are dogs, cats and parrots andfor that reason they are included.

The example shows a relevant aspect to manage in problems expressed in natural lan-guage: contextual and implicit information must be incorporated to the syllogism. Thosesyllogistic patterns limited to predefined patterns or with a small number of premises cannotmanage this kind of problem.


PR1 : At least 70% of sixth course students passed PhysicsPR2 : At least 75% of sixth course students passed MathematicsPR3 : At least 90% of sixth course students passed PhilosophyPR4 : At least 85% of sixth course students passed Foreign Language

C : QC sixth course students passed Physics, Mathematics, Philosophyand Foreign Language

Table 2.11: Syllogism with five terms and five premises.

2.3.1.2 Example 2: Sixth Course Students

The following example is mainly focused on the use of interval quantifiers. The exercise isextracted from the Spanish Mathematical Olympiad 1969-702:

In the tests of sixth course in a secondary school, at least 70% students passed thesubject of Physics; at least 75% students passed Mathematics; at least 90% passedPhilosophy and at least 85% passed Foreign Language. How many students, atleast, did pass these subjects?

In this case, five terms must be considered in the universe E: students in the sixth course

= P1, students that passed Physics = P2, students that passed Mathematics = P3, students

that passed Philosophy = P4 and students that passed Foreign Language = P5. Table 2.11shows a possible formalization, where the corresponding quantifiers are: Q1 = [0.7,1] ofPR1; Q2 = [0.75,1] of PR2; Q3 = [0.9,1] of PR3; Q4 = [0.85,1] of PR4; and QC = [a,b] thequantifier of the conclusion.

In this case, we avoid the details regarding the corresponding set of inequations of eachpremise. According to section 2.2.2, 25 = 32 disjoint sets must be generated and the corre-sponding inequations. The quantifier of the conclusion is the binary proportional type and itmust be calculated by applying linear fractional programming obtaining QC = [0.2,1]; i.e.“at least 20% of sixth course students passed Physics, Mathematics, Philosophy and ForeignLanguage”, which corresponds to the expected result.

2http://platea.pntic.mec.es/csanchez/olimp_1963-2004/OME2004.pdf ; p. 29


2.3.2 Syllogisms with Fuzzy Quantifiers Approximated as Pairs of Inter-vals

In this method, each fuzzy quantifier Q in the syllogism is defined as Q = {KERQ,SUPQ},where KERQ = [b,c] corresponds with the kernel and SUPQ = [a,d] corresponds with thesupport. In this model, the solution is obtained from two systems of inequations, the firstof them taking KERQ as crisp interval definitions for all the statements in the syllogism andthe second one taking SUPQ. The corresponding solutions define KERQC and SUPQC for thequantifier QC, respectively. We should note that this is an exact and very simple approachfor cases where the quantifiers are trapezoids. For the cases of non-trapezoidal quantifiersthis produces an approximate solution. In the cases where non-normalized quantifiers mayappear (either in the definition of the premises or due to the non-existence of solutions for theinequations system) this approach cannot produce results and therefore solutions should beobtained by using the approach described in section 2.3.3.

For the example in Table 2.11 we have the following quantifiers: Q1 = {[0.8,0.9], [0.7,1]};Q2 ={[0.8,0.85], [0.75,0.9]}; Q3 = {[0.92,1], [0.9,1]}; Q4 = {[0.9,0.95], [0.85,1]}; andQC = {[b,c], [a,d]} the quantifier of the conclusion.

Applying the procedure described in section 2.3.1 for each system, we obtain KERQC =

[0.42,1] for the system defined by taking the Kernels of all the quantifiers in the premisesand SUPQC = [0.20,1] for the system defined by taking the Supports of all the quantifiers inthe premises. The result is therefore QC = {[0.42,1], [0.2,1]} which is a result that entailsthe definition for QC = “At least 20% by mainly from 42%” obtained in the previous section.Figure 2.2 shows the graphical representation of the quantifier QC, where the interval [0.2,1]denotes the support of the fuzzy set and the interval [0.42,1] denotes its kernel.

As we can see, by using this approach we can only calculate the two represented intervals.Any other value between them must be interpolated. The main problem of this approach isfor fuzzy quantifiers that are not represented as trapezoidal functions or those that are non-normalized. In these cases, we must use fuzzy quantifiers.

2.3.3 Syllogisms with Fuzzy Quantifiers

The use of fuzzy quantifiers supposes a generalization of the reasoning procedure describedin the previous sections. Each fuzzy quantifier is managed through a number of α−cuts,which are crisp intervals defined on the universe of discourse of the quantifier. Therefore, an


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000

0,2

0,4

0,6

0,8

1

Percentage %

[0.42,1]

[0.20,1]

Figure 2.2: Graphical representation of the solution obtained for QC for the example inTable 2.11, SUPQC = [0.2,1] and KERQC = [0.42,1].

inequation system similar to the ones in section 2.2.3 is obtained for each α−cut considered.The reasoning procedure consists of applying the method described in section 2.2.4 for eachof these inequation systems. Each of the solutions obtained for all these systems define thecorresponding α−cut for the quantifier QC in the conclusion. By using this general approach,it is possible to manage any generalized quantifier (as the ones shown in Table 2.1) for threesituations that are not considered in the previous models:

– trapezoidal quantifiers where non-normalized results are obtained in any of the solu-tions, since for this case KERQC cannot be calculated and therefore the previous ap-proach leads to indefinition.

– quantifiers defined with non-trapezoidal functions (in this case, this is an approximatesolution where the level of approximation can be defined as we wish) and

– typical linguistic fuzzy quantifiers like “most”, “many”, “all but around three”, etc.

In order to better illustrate this approach we shall consider the five types of examplesdescribed in the subsequent sections.


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Percentage %

[0.20,1]

[0.22,1]

[0.24,1]

[0.26,1]

[0.28,1]

[0.30,1]

[0.33,1]

[0.35,1]

[0.20,1]

[0.39,1]

[0.42,1]

Figure 2.3: Graphical representation of QC with 11 α−cuts.

2.3.3.1 Example 1 (Fuzzy Extension): Students of Sixth Course

This first example is the one in Table 2.11 which is useful for showing the consistency of thisapproach with the one described in the previous section. We shall consider fuzzy trapezoidalquantifiers that comprise the definitions for the quantifiers in Table 2.11. Using the usual nota-tion for trapezoidal fuzzy membership functions where Q= [a,b,c,d] (KERQ = [b,c],SUPQ =

[a,d]) we have: Q1 = [0.7,0.8,0.9,1]; Q2 = [0.75,0.8,0.85,0.9]; Q3 = [0.9,0.92,1,1]; Q4 =

[0.85,0.9,0.95,1]; and QC = [a,b,c,d] the quantifier of the conclusion. Applying the corre-sponding optimization method, we obtain QC = [0.2,0.42,1,1], which is consistent with theresults in the previous sections. Figure 2.3 shows a graphical representation of QC for thecase considered involving eleven α-cuts. As we can see, the eleven α-cut intervals allow usto better approximate a fuzzy definition for the conclusion quantifier QC. It is worth notingthat the extreme cases, α−cut= 0 and α−cut= 1, correspond to the support and kernel casesin the approximation described in the previous section 2.3.2 (see Figure 2.2).

2.3.3.2 Example 2 (Non-normalized Fuzzy Extension): Students of Sixth Course

We now illustrate a syllogism that produces a non-normalized fuzzy set as result. We consideragain the example of Table 2.11 but adding the following premise:


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Percentage %

[ 0.20,0.60 ]

[ 0.22,0.58 ]

[ 0.24,0.56 ]

[ 0.26,0.54 ]

[ 0.29,0.52 ]

[ 0.31,0.50 ]

[ 0.33,0.48 ]

[ 0.35,0.46 ]

[ ]0.38,0.44

[ ]0.42,0.42[0.40,0.40]

Figure 2.4: Graphical representation of the solution obtained for QC for the example in Ta-ble 2.11 with the additional premise PR4.

PR5 : Between 40% and 90% sixth course students did not pass Physics or Mathematics orPhilosophy or Foreign Language

Its definition is Q5 = [0.4,0.6,0.8,0.9] and the corresponding system of inequations isadded to the previous system. Results are shown in Figure 2.4. For α−cuts higher than 0.95the system has no solution; i.e., the fuzzy set that defines the quantifier of the conclusion isa non-normalized fuzzy set. Therefore, this syllogism cannot be solved by managing fuzzyquantifiers approximated as pairs of intervals (section 2.3.2).

2.3.3.3 Example 3 (Non-trapezoidal Result): Wine Warehouse

In this case, we show a syllogism using the class of proportional quantifiers known as RegularIncreasing Monotone (RIM) quantifiers [89], that were proposed within the framework ofquantifier guided aggregation of combination of criteria. This is a way of defining proportionalquantifiers (such as “few”, “many”, “most”, etc.) where the linguistic term is interpreted as afuzzy subset Q of the [0,1] interval. Its basic definition is shown in equation 2.14

Qα(p) = pα ;α > 0 (2.14)

where Qα denotes a linguistic quantifier and p ∈ [0,1]. For instance, for Q2 (labelled as“most”), Q2(0.95) = 1 means that saying 95% completely fulfills the meaning conveyed by


PR1 : Q1 bottles of red wine are sold in the United KingdomPR2 : Q1 bottles of red wine sold in the United Kingdom

are bought by J. MoriartyC : QC bottles of red wine are sold in the United Kingdom

and bought by J. Moriarty

Table 2.12: Zadeh’s Intersection/product syllogism.

“most” and Q2(0.6) = 0.75 means that saying 60% fulfils with degree 0.75 the meaning con-veyed by “most”. Other usual RIM quantifiers are Q0.5, usually labelled [34] as “a few” andQ1 (identity quantifier, also labelled as “a half”). RIM quantifiers are also associated with thesemantics of “the greater the proportion of. . . the better” owing to its increasing monotonicbehaviour [89]. Under this interpretation, the semantics should be labelled accordingly to theactual value of α as “linear” (α = 1), “quadratical” (α = 2), “sub-linear” (α < 1), etc.

So, let us consider the following example using RIM quantifiers: in a wine warehousethat sells red wine, white wine and other products derived from grapes, we have the followingsentences:

Q1 bottles of red wine are sold in the United Kingdom.Q1 bottles of red wine sold in the United Kingdom are bought by J. Moriarty.

Both propositions involve the identity quantifier Q1 (α = 1), that can either be interpretedas “a half” or as “the greater. . . the better” (linear, α = 1).

If we apply the Zadeh’s [93] Intersection/product pattern, we can infer the following state-ment: “QC bottles of red wine are sold in United Kingdom and bought by J. Moriarty”. So,the corresponding syllogism is shown in Table 2.12.

As the result of the corresponding optimization method, we obtain the quantifier QC shownin Figure 2.5, such that QC = Q0.5. Its linguistic interpretations are “a few” [34] “a fewbottles of red wine sold in the United Kingdom are bought by J. Moriarty” or “the greaterthe proportion of bottles of red wine that are sold in the United Kingdom and bought by J.Moriarty the better” (sublinear, α = 0.5).

In case this example was solved using the KER− SUP linear interpolation approach amuch worse approximation (with bigger error) would be obtained for the inferred sublinearquantifier QC = Q0.5 as it is shown in Figure 2.5.


[ 0,1 ]

10.90.80.70.60.50.40.30.20.1

0[ 0.01,1 ]

[ 0.04,1 ]

[ 0.09,1 ]

[ 0.16,1 ]

[ 0.25,1 ]

[ 0.36,1 ]

[ 0.49,1 ]

[ 0.64,1 ]

[ 0.81,1 ]

[1,1]

0 0.5 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Figure 2.5: Qc of Example 2.12.

PR1 : Many sales are from red winePR2 : Few sales are from white wine

C : QC sales are not from red wine or white wine

Table 2.13: Zadeh’s Consequent Disjunction syllogism with different conclusion.

2.3.3.4 Example 4 (proportional fuzzy quantifiers): Account of a Wine Ware-house

In this case, we show a syllogism involving fuzzy quantifiers of the type “many”, “most”, etc.Let us consider the account of the wine warehouse, that is not so good, and they only managethe following data:

Many sales are of red wine. A few sales are of white wine.

Taking both statements as premises in the universe of products sold by wine warehouse =

E, we have a syllogism with three terms: sales=P1, red wine=P2 and white wine=P3. Withthese two premises we can use, for example, consequent disjunction pattern of Zadeh [93],the conclusion of which takes the form QC P1 are P2 or P3 with P1 being the subject of thepremises and P2 and P3 the corresponding predicates. However, in this example, we changethe conclusion for the following one “QC sales are not of red wine or white wine”, because thisallows them to know how many sales are of other products. Table 2.13 shows the completesyllogism.


Percentage %

10.90.80.70.60.50.40.30.20.1

00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Figure 2.6: Graphical representation of the solution obtained for QC for the example in Ta-ble 2.13.

The first step to perform in the inference process is to assign the corresponding trapezoidalfunction to each one of the linguistic quantifiers of the premises. On the basis of [76], weassign the following trapezoids: many = [0.5,0.55,0.65,0.7] and f ew = [0.1,0.15,0.2,0.25].Next, we apply the reasoning procedure obtaining the following result: QC = [0,0,0.45,0.5];this can be associated with the following semantics: “at most half”; therefore, the conclusioncan be stated as: “at most half sales are not of red wine or white wine”. In Figure 2.6 weshow the corresponding fuzzy set. It is relevant to note that the obtained QC of this exampleis a decreasing quantifier, a type of quantifier that some classical models of fuzzy syllogismcannot manage [93, 57].

The example shown in Table 2.13 shares the same aspect of P. Peterson’s IntermediateSyllogisms [63], but with a different conclusion, as Peterson’s schemas only include simpleconclusions, without involving logical operations such as disjunction. Table 2.14 shows thesyllogism of Table 2.13, but with different terms for a more clear example (people, white

hats and red ties) and according to BKO (B meant that PR1 involves the quantifier “few”; Kthat PR2 involves the quantifier “many”, and O that C involves the quantifier “some. . . not” )Peterson’s schema of Figure III. Using the same previous definitions for the “few” and “many”quantifiers, we obtain the following result: QC = [0,0,0.98,0.98], which is consistent with theusual definition for the quantifier some. . . not= [0,0,1− ε,1− ε].


PR1 : Few people are wearing white hatsPR2 : Many people are wearing red ties

C : QC people that are wearing red ties do not wear white hats

Table 2.14: Peterson’s BKO syllogism.

PR1 : All but around fifteen boxes of wine are for J. MoriartyPR2 : Around four boxes of the boxes that are not for J. Moriarty

are for J. WatsonC : QC boxes are not for J. Moriarty neither for J. Watson

Table 2.15: Zadeh’s Intersection/product syllogism.

2.3.3.5 Example 5 (Exception Fuzzy Quantifiers): Wine Warehouse

The final example shows a syllogism combining fuzzy quantifiers that have not been previ-ously considered in the literature as exception quantifiers (i.e., “all but around four”, etc.)combined with absolute ones (i.e., “around four”, etc.):

All but around fifteen boxes of wine are for J. Moriarty. Around four boxes ofthe boxes that are not for J. Moriarty are for J. Watson.

We can identify the following three terms in the statements: boxes of wine = P1, boxes

for J. Moriarty = P2 and boxes for J. Watson = P3. Given this information, we can infer howmany boxes are not for J. Moriarty neither for J. Watson; that is, “QC boxes are not for J.Moriarty neither for J. Watson”. In Table 2.15 is afforded the complete argument.

The quantifier of PR1 is a fuzzy quantifier of exception, as it indicates that there are around

fifteen boxes of wine that are not for J. Moriarty. For instance, the associated trapezoidalfunction to this number can be around 15= [13,14,16,17]. The quantifier of PR2 is anotherfuzzy quantifier, but in this case an absolute one. This kind of quantifier has been analysedin the literature from the very beginning (Zadeh’s distinction between absolute/proportionalfuzzy quantifiers [90]), but it has not been considered from the point of view of its use inreasoning. In this case, we assign around four= [3,4,4,5].

After applying the reasoning process, we obtain the absolute quantifier QC = [8,10,12,14]for the conclusion, with the usual associated meaning of “around eleven” (since value 11 isin the middle of the trapezoid’s kernel). Therefore, in this example the conclusion states that“around eleven boxes are not for J. Moriarty neither for J. Watson”.

2.4. Combination of Absolute and Proportional Quantifiers 125

On the other hand, it is worth noting that this model is consistent with a fuzzy arithmeticapproach to this problem; since from PR1 we have that “around fifteen’ boxes are not for J.Moriarty, and from PR2 we have that “around four” of these are not for J. Watson; therefore,“around fifteen” “around four” = “around eleven”, which is the number of boxes that arenot for J. Moriarty neither for J. Watson.

2.4 Combination of Absolute and Proportional Quantifiers

In the analyses of the most relevant syllogistic patterns of the literature presented in chapter 1we concluded that only in Exception Syllogistics was the possibility of combination propor-tional and absolute quantifiers (exception ones) addressed. However, this proposal only con-sidered the combination of quantifiers in the same statement for determining the value of thesubject-term and, therefore, the inference mechanism did not have any relevant innovation.

However, in natural language it is not difficult to think of in arguments involving state-ments with different type of quantifiers. For instance, “there are around forty students in thisclassroom, most of them are blond; therefore, no more than five or six are not blond”. Al-though this is an example of fuzzy syllogism and the quantifier of the conclusion (“no morethan five or six”) may be a matter of debate, nobody would say that this is a fully invalid infer-ence. Therefore, the combination of absolute and proportional quantifiers in a same syllogismcan be easily exemplified in natural language and, therefore, it constitutes a challenge for anynew proposal of syllogistic reasoning system.

The difference between an absolute and proportional reading of a linguistic quantifier is atopic that has ben deeply analysed in linguistics. In [44, 21] and interesting analysis of thisquestion is described. Both works analyse the case of the quantifier “many” and match in theirdefinitions for the absolute and proportional readings. So, let us consider a proposition many

S are P:

– Absolute (or cardinal) reading: |S∩P|> n; n a real number.

– Proportional reading: |S∩P||S| ≥ k; k a fraction or percentage.

Given that “many” is a vague quantifier, the values of n and k are context-dependent.Other quantifiers, such as “few”, “many”, etc. also have both possibilities of reading and intheir definitions > and ≥ can change for < and ≤.


PR1 : All kings are powerfulPR2 : Some kings are single

C : QC powerful people are single

Table 2.16: AII mood of Figure III.

Exception Syllogistics (section 1.1.3), the only syllogistic system that deals with argu-ments with absolute quantifiers, pointed out that a value for S must be assigned to a right in-ference and if it is unknown, it should be estimated. Therefore, this idea must be incorporatedto our model. It is flexible enough and there is no a priori problem in building inequationssystems that involve simultaneously fractional and linear equations as they can be solved us-ing fractional programming techniques. From a computational point of view, the key point isto assign a large enough size to the universe of discourse. Thus, any more or less large valuefor n but also expressed in terms of percentage to satisfy k can be used in the syllogism.

To illustrate this idea and, for the sake of simplicity, we shall consider the AII mood(Figure III) to show how the combination between the proportional and absolute quantifierswork; since A involves the proportional quantifier “all” and I the absolute one “some”. Thisis shown in Table 2.16.

The conclusion of this argument is also an I statement. Hence, let us consider Power f ul =

P1, King = P2 and Single = P3 (see Figure 2.1). To preserve the Aristotelian meaning, weshall use crisp definition for these quantifiers according to Table 2.8 (left); i.e., all = 100%and some≥ 1. According to Table 2.8 (right), the corresponding system of inequations is:

PR1 :x7 + x8

x5 + x6 + x7 + x8= 10100;

PR2 :x4 + x8 ≥ 1;

C :x6 + x8 ≥ a;

(2.15)

As can be observed, the right-hand member of the equation PR1 is a large enough valuebut preserving the proportional meaning of “all”, as it still denotes the inclusion of the subject-term in the predicate-term, and it avoids conflicts with PR2. Applying the fractional program-ming techniques, we obtain QC ≥ 1, the right result; i.e., “some powerful people are single”.

2.5. Summary 127

2.5 Summary

In this chapter we have described our proposal for a syllogistics that focused on improving theweaknesses of the models analysed in the previous chapter; i.e.: i) non-fuzzy approaches thatinvolve vague quantifiers have a logic foundation rather a heuristic one, generating a relevantamount of new patterns but are too rigid; ii) fuzzy approaches propose heuristic proceduresfor calculating the conclusion of the argument in terms of the quantifiers of the conclusion,but they are not fully compatible with Aristotelian Syllogistics; iii) many of the new proposedpatterns are practically limited to the form of three terms, two premises and a conclusion. Theremaining ones only consider some additional premises but the same number of patterns; and,iv) only schemas with absolute (such as exception ones) or proportional quantifiers (such asproportional ones) are considered. There is no model that proposes how to combine themalthough this is a common situation in natural language.

Thus, our approach consists of formulating a new framework of syllogistic reasoning,firstly, compatible with TGQ, the standard theory of quantification in natural language. Hence,preserving the standard form of quantified statements (Q S are P), we introduce new quanti-fiers into syllogism, such as comparative or exception ones, under the same approach. Further-more, this framework also allows us to use crisp, interval and fuzzy definitions of quantifierswithout changing the procedure.

On the other hand, we assume the heuristic strategy, emphasized by Probabilistic Syllo-gistics, and adopt an algebraic theory of reasoning, as in Interval Syllogistics, but supportedon Venn diagrams, as Interval and Exception Syllogistics. Thus, we transform a reasoningproblem into a mathematical optimization problem, where the quantifier of the conclusion iscalculated in terms of the quantifiers of the premises. Hence, from each premise n inequationsare generated, depending on the definition of the quantifier (crisp, interval or fuzzy), whichare the constraints for optimizing the conclusion, which is also expressed as inequations. Thetechniques used to deal with with operations are Simplex and fractional programming.

We also consider a fundamental challenge to our proposal to be compatible with a lin-guistic version of syllogism, where the quantifiers can be ordered into an LSO or anotherstructure, such as in Interval Syllogistics, where each quantifier have associated a crisp value,an interval or a fuzzy set and the result of the inference procedure is approximated to thecloser definition.


This model is partially implemented in a software library called SEREA (Syllogistic Epis-temic REAsoner) that is capable of managing the reasoning scheme described in this paper.The current version of SEREA is freely downloadable from our website [2].

Having analysed the case of vagueness in quantifiers, the case of vagueness in terms alsodeserves our attention, as this is also a very common phenomenon in natural language. So,how can a syllogism that involves vague terms as middle term be qualified? Can a valid oracceptable syllogism be obtained or only a fallacy? In the next chapter, we shall explore thisidea. This leads us to a new form of syllogism where the problem is not the calculation of thequantifier of the conclusion in terms of the quantifiers of the premises, but the reliability of orconfidence in the conclusion given the approximate character of the chaining.

CHAPTER 3

APPROXIMATE CHAINING SYLLOGISM

In the two previous chapters, we described different approaches to syllogism where vaguenessonly appears in the quantifiers. However, as stated at the end of chapter 2, vagueness can alsoappear in the terms because they are natural language words. As have previously stated,syllogism is a form of chaining reasoning and, therefore, vagueness in the middle term is themost interesting case for producing a new form of reasoning pattern not considered until nowand for reconsidering some arguments usually qualified as fallacies; the so-called fallacy of

four terms and the fallacy of ambiguous middle term [12]. The fallacy of four terms appearswhen the syllogism involves two different terms in the middle term (DT); for instance, “allstudents are tall, all dogs are pets; therefore, all dogs are tall” (AAA-Figure I) is fallaciousbecause students and pets are different. The fallacy of the ambiguous middle term appearswhen different meanings of the middle term are used in the different premises; for instance,“all revolutionary people have brilliant ideas, all anarchist are revolutionary; therefore, allanarchist have brilliant ideas” (AAA-Figure I) is fallacious as in the first premise revolutionary

refers to those people that defend the abolition of government, while in the second premise itrefers to people whose have innovative ideas.

Nevertheless, in certain cases, these arguments are not fallacies but approximate infer-ences. Let us consider the example shown in Table 3.1. The DT is played by two differentterms, disaster and calamity, and, therefore, it falls into the fallacy of four terms. Neverthe-less, from a pragmatic or everyday reasoning point of view, it is an approximate but acceptableinference as the speakers know that the terms disaster and calamity are similar enough to pre-serve a chaining that is sufficiently strong. In this case, any weakness point in the chaining

130 Chapter 3. Approximate Chaining Syllogism

PR1: All earthquakes are a disasterPR2: No calamity is desirable

C: No earthquake is desirable

Table 3.1: Syllogism with vagueness in the DT.

affects the reliability of the syllogism, which should be qualified in terms of acceptabilityrather than validity.

In this chapter we shall focus on this type of syllogism. Our motivation is because, al-though the chaining can be constituted by two different terms, they can have a relationshipof similarity or a family resemblance between them that is sufficiently strong to render theargument acceptable for the speakers. To qualify the reliability of the argument, we proposeto evaluate the conclusion in terms of truth value or degree of confidence, which depends onthe strength of the chaining between the terms. We assume the following statement as anaxiom: the greater the similarity between the terms, the higher the degree of confidence in the

conclusion.The best way, from our point of view, to measure this eventual vague link is in terms of

similarity, as this is a well defined concept both in maths and fuzzy logic. We identify threepossible situations: i) similarity in terms of fuzzy inclusion, ii) similarity in terms of fuzzyoverlapping, and, iii) similarity in terms of synonymy. The first two have been studied indepth in fuzzy logic and we only make a brief description of them; case iii), although a verytypical and intuitive semantic relationship in natural language, is less studied in this field.

This chapter is organized as follows: in section 3.1 different proposals for dealing withsimilarity between terms are analysed; in section 3.2 we elaborate a measure of synonymybetween two terms; in section 3.3, we propose a basic inference pattern for this type of syllo-gisms including some examples; and, finally, in section 3.4, we summarize the main contri-butions of this chapter.

3.1 Similarity between Terms: Fuzzy Inclusion, Fuzzy Overlap-ping and Synonymy

We address the syllogism with fuzzy terms (which we shall call Approximate Chaining Syl-logism (ACS)) using the set-based interpretation of quantified statements, since it is the oneused in our proposal of syllogistic system (see chapter 2). As we said above, we identify

3.1. Similarity as Fuzzy Inclusion, Fuzzy Overlapping and Synonymy 131

DT

MTNT DT'

Figure 3.1: Counter example of ACS when DT ′ is included in DT .

three possible scenarios with three different underlying ideas of similarity: i) fuzzy inclusion,ii) fuzzy overlapping and, iii) synonymy. In the subsequent sections, each one them will beanalysed.

3.1.1 Similarity as Fuzzy Inclusion

Fuzzy inclusion is a type of operation well known in fuzzy logic. It denotes the degree towhich a fuzzy set is a subset of another fuzzy set such as, for instance, the inclusion of thefuzzy set slippery roads in the fuzzy set of dangerous roads. There are different definitions;for instance, let A and B be two (fuzzy) sets in a universe U with B⊆ A;

µB(ui)≤ µA(ui) ; ∀ui ∈U (3.1)

or

µB(ui)≤ µA(ui) ; B⊆ A≡ B→ A (3.2)

For the ACS, fuzzy inclusion is valid in a particular case; i.e., when the middle term of theMinor Premise is a (fuzzy) subset of the middle term of the Major Premise (see Figure 3.2).In the opposite direction, when the middle term of the Major Premise is a (fuzzy) subset ofthe middle term of the Minor Premise, is not valid as a simple and clear counter example canbe generated, as Figure 3.1 shows, where DT denotes the middle term of the Minor Premiseand DT ′ the middle term of the Major one, NT the minor term and MT the major term.

For the right direction of the inclusion, a counter example can also be generated, but, inthis case, it is necessary to assume as a precondition a substantial difference in the size of thesets or a degree of fuzzy inclusion very low, as can be deduced observing Figures 3.1 and 3.2.


MT DT'NT

DT

Figure 3.2: Counter example of ACS when DT is included in DT ′.

PR1 Some people are intelligentPR2 No wise person ignores E = mc2

C Some people do not ignore E = mc2

Table 3.2: Syllogism with approximate chaining.

PR1 Some people are intelligentPR2 No wise person ignores E = mc2

PR3 Almost all wise people are intelligentC Some people do not ignore E = mc2

Table 3.3: Syllogism where the additional premise for Spanish and European.

Table 3.2 shows an example of syllogistic argument involving the terms intelligent andwise in the chaining. The users assume that wise people ⊆ intelligent and, thus, they can belinked trough a fuzzy inclusion.

Other possibility to make explicit the inclusion of DT in DT’ is to introduce an additionalpremise in the syllogism with the form “almost all wise people are intelligent” and generate apoly-syllogism [12], as it is shown in Table 3.3.

3.1.2 Similarity as Overlapping

Another possible relationship of similarity between two terms A and B is the so-called fuzzyoverlapping. This idea of chaining was initially proposed in Mamdani’s reasoning style [10]and subsequently extended to other type of fuzzy reasoning [11]. Although Mamdani’s rea-soning style is different to syllogism, the underlying idea can be applied to it. For a better

3.1. Similarity as Fuzzy Inclusion, Fuzzy Overlapping and Synonymy 133

Set of premises If X is A, then Y is BIf Y is B’, then Z is C

Observation X is A’Conclusion Z is C’

Table 3.4: Mamdani’s Approximated Chaining schema.

0

1A A'

Figure 3.3: α indicates the overlapping between A and A′.

explanation of this idea, we shall explain the concept of fuzzy overlapping using a Mamdani’sargument and other applying fuzzy overlapping to the syllogism.

Arguments in Mamdani’s reasoning involves a set of rules, an observation and a conclu-

sion. The rules take the form “If X is A then Y is B”. The chaining is through the antecedentand the consequent parts: there is an intermediate variable in the consequent part of a rule (orsome rules) and in the antecedent part of other(s) [10]. When a relaxation in the chaining isassumed, the pattern shown in Table 3.4 is produced. Thus, the result of the consequent ofthe first premise (Y is B) must be reassessed when is the antecedent of the second premise(Y is B’). One way to do so is to measure the overlapping between B and B′ (see Figure 3.3)obtaining the degree of fulfilment (α) associated to it.

This idea can be easily adapted to ACS, whether the middle term comprises two termswhere one of them involves a linguistic modifier of the other, or whether they are differentlabels in the same linguistic partition. Let us consider the example shown in Table 3.5, wherewet (DT) and extremely wet (DT’) are the middle terms. If a linguistic partition for evaluatinghumidity is considered, wet and extremely wet are two labels with certain degree of overlap-


Some spring days are wetAll extremely wet days are cloudySome spring days are cloudy

Table 3.5: Syllogism with overlapping between DT and DT’.

All earthquakes are a disasterNo calamity is desirableNo earthquake is desirable

Table 3.6: Syllogism with DT and DT’ synonyms.

ping, which affect to the degree of confidence of the conclusion: the higher the degree ofoverlapping, the higher the confidence on the conclusion.

3.1.3 Similarity as Synonymy

The last type of relationship of similarity that we propose is synonymy, a relationship thatdenotes resemblance among the meanings of words. It is a very common and intuitive forspeakers in natural language but the modelling thereof is not obvious in terms of fuzzy sets.Let us consider again the example of the introduction (Table 3.6), where the middle terms aredisaster (DT) and calamity (DT’).

We cannot properly say that disaster and calamity have a inclusion relationship as neitheris disaster a subset of calamity nor calamity a subset of disaster. Nor are they labels on alinguistic partition on the universe of bad events, where neither of them can be built througha linguistic modifier or a shifting over the other; for instance, calamity are not, for instance,serious disasters or more negative than disasters. Hence, neither of the two previous cases canbe applied. However, as we have already stated, this is an acceptable syllogism as the conclu-sion follows reasonably from the premises given that the chaining is established through thestrong link between disaster and calamity.

In the next section, we shall analyse in depth the concept of synonymy and will propose ameasure to calculate a degree of synonymy between two terms.

3.2. A Proposal for a Measurement of Synonymy between Two Terms 135

(a) Syllogism AAA - Figure I.

PR1: All human beings are mortalPR2: All Greeks are human beings

C: All Greeks are mortal

(b) Syllogism AAA - Figure I with DT interchanged.

PR1: All featherless bipeds are mortalPR2: All Greeks are featherless bipeds

C: All Greeks are mortal

Table 3.7: Examples of syllogisms with the terms interchanged.

3.2 A Proposal for a Measurement of Synonymy between TwoTerms

We address synonymy between terms attending to the aim of this research: evaluating thevalidity of a syllogism where the middle term is played by two synonymous terms. Thissubstitution can be performed in two senses: i) changing all the occurrences of the DT by asynonym; ii) changing only some occurrences of the DT for the synonymous term.

For case i), let us consider the argument of Table 3.7(a). The role of DT is played by theterm human being; if we change this term for featherless bipeds, Plato’s definition for human

beings, we obtain a new syllogism (see Table 3.7(b)) equivalent to the previous one becauseboth terms are completely equivalent.

This case is not the most relevant one as if we substitute all the occurrences of the DT bya synonym, we obtain an equivalent syllogism preserving the precise chaining.

Case ii) entails, in fact, a modification in the chaining and we shall focus on it. So, let usconsider the example shown in Table 3.6. The role of DT is played by the terms disaster andcalamity. To preserve the validity of the argument, the terms must be or totally equivalent or,at least, they must have a very similar meaning. In this context, they satisfy the condition orhaving a certain degree of similarity and, hence, the argument is acceptable.

Nevertheless, the definition of synonymy and what type of relationship it denotes is stilla question with multiple dimensions. In the literature, we can find two main approaches tosynonymy [74]:

1. Synonymy as a theoretical challenge: It is a theoretical problem and, therefore, its mainaim is to obtain a clear definition of it. Thus, synonymy is understood as a preciserelationship and its criterion is identical meaning.

2. Synonymy as a phenomenon of natural language: It is studied from the use of thespeakers; therefore, it is a experimental challenge. Thus, synonymy is understood a


relationship of similarity, not identity, whose criterion is similar meaning according tothe context of utterance.

Defenders of the first perspective can be found in the field of philosophy of language.For instance, W. O. Quine advocates this point of view. Thus, if we consider modelling syn-onymy as a crisp relation, we must identify meanings with equivalence classes of synonymousexpressions [66]; in [67, p. 32], he said:

A truth that can be turned into a logical truth by substituting synonyms for syn-onyms. Statements may be said simply to be cognitively synonymous when theirbiconditional (the result of joining them by “if and only if”) is analytic.

If we assume this definition of synonymy, any syllogism involving two synonyms in themiddle term is a full valid syllogism as the terms are equivalent from this perspective.

The second perspective also has numerous defenders, starting from Gabriel Girard [24].He made an approach to this idea stating that:

In order to obtain property, one does not have to be demanding with words; onedoes not have to imagine all that so-called synonyms are so with all the rigor-ousness of perfect resemblance; since this only consists of a principal idea whichthey all enunciate, rather each one is made different in its own way by an acces-sorial idea which gives it its own singular character. The similarity bought aboutby the principal idea thus makes the words synonyms; the difference that stemsfrom the particular idea, which accompanies the general one, means that they arenot perfectly so, and that they can be distinguished in the same manner as thedifferent shades of the same colour.

Currently, although both perspectives are valid, the most widely accepted is the secondone; i.e., a relationship of similarity that must be studied from its empirical use. We adopt theframework proposed in [74, 75], where a pragmatical study of synonymy using a thesaurus1

is proposed as it gathers the use of synonyms of the speakers of a language. Furthermore, twoadditional reflections about synonymy are assumed:

1. It has a gradual character. Thus, although two expressions can be synonym to a certaindegree, this degree may not be high enough for making them interchangeable. So, for

1Thesaurus is a book that lists words in groups of synonyms and related concepts.


instance, car and automobile are interchangeable in an informal conversation betweentwo people about buying a new car; they are not interchangeable in a report about trafficaccidents because automobile not only refers to car accidents but also motorbike or lorryaccidents.

2. The polysemy of the terms. Sometimes, two synonymous terms cannot be interchange-able as one of them is polysemic and the meaning in a particular statement does notcorrespond with the synonym. For instance, the polysemic word table is synonymousof desk in one of its meanings; however, it cannot be substituted by this one in thestatement “the data in the table of your last paper is wrong”.

As we have stated previously, an experimental criterion of synonymy is based on whatwords the speakers of a language actually consider as synonyms. Thesaurus dictionaries arethe documents that gather this information. Thus, as criterion of synonymy, we adopt the fol-lowing: two words are synonyms if both words appear as synonyms in a thesaurus dictionary.Nevertheless, a couple of additional conditions must be taken into account:

1. Natural languages are systems in constant evolution and, accordingly, synonymy, thatis characterized by its graduability and its context-dependence, also changes constantly.Hence, the thesaurus must be improved according to the changes in the correspondinglanguage.

2. It is reasonable to think that synonymy is a symmetric property (if a is synonymous withb then b is synonymous with a), but this is not always the case in a thesaurus dictio-nary. The usual origin of this situation is that the relationship between the terms is notexactly synonymy but relationships of different nature such as hyponymy, hiperonymy,meronymy, etc., which are not symmetric.

3. With respect to transitive property, if a is synonymous with b and b is synonymous withc, a is synonymous with c, this may not be the case. For instance, we can say thatcar is synonymous of automobile and automobile is a synonym of motorbike but car

is not synonymous with motorbike. From the point of view of reasoning, this questionmust be taken into account for the case of polyllogisms (syllogisms with more thantwo premises), as the chaining can be broken by an inadequate transitivity between theterms.


Thus, we shall adopt an experimental definition of synonymy based on a thesaurus. Thisleads us to interpreting the synonymy between two terms as a question of degree. Therefore,the next step is to define a procedure to measure or calculate the similarity, resemblance orsynonymy between two terms. This question will be addressed in the next section.

3.2.1 Calculation of the Degree of Synonymy

Following [74, 75], we adopt a framework where the degree of synonymy between two termsis expressed as x, where x ∈ [0,1] according to the information retrieved from a thesaurusdictionary. For instance, for two given terms a and b, it is possible to calculate their degreeof synonymy x. The elaboration of this measure is divided into two steps: i) selection of athesaurus dictionary; ii) model for calculating degrees of synonym.

We adopt an on-line thesaurus dictionary as they usually are in constant evolution andincorporate the changes in the uses of the speakers in their language better (one of our pre-requisites) and consulting them is also very easy in general. On the other hand, they usuallyinvolve both conceptual-semantic and lexical relations, generating a wide set of relationshipsamong terms, which satisfies the other two prerequisites that we have established: to distin-guish among the subtypes of synonymy (hyponymy, hiperonymy, etc.) that are not symmetricand to consider the transitivity problem. We shall use WordNet [35] to illustrate our examples.

WordNet is a “large lexical database of English. It includes nouns, verbs, adjectives andadverbs that are grouped into sets of cognitive synonyms (synsets), each expressing a distinctconcept” [1]. Thus, each concept provides two types of information: synset and gloss. Synsetis the class terms that have synonymy relationship (total or partial) of a particular concept;each one of the synonyms that compound a synset is called a variant. Gloss is a distinctivedefinition for characterizing a synset. Figure 3.4 shows a screen capture using the term write.

It is a polysemic word and, hence, it has various glosses that appear in brackets:

– produce a literary work

– communicate or express by writing

– write music

– mark or trace on a surface

– record data on a computer


Figure 3.4: The term write on WordNet.

– write or name the letters that comprise the conventionally accepted form of (a word orpart of a word)

– create code, write a computer program

The glosses are not defined using an univocal way but they can adopt different forms.In [26, p. 64-65] nine different patterns for defining glosses are identified.

Regarding the synsets associated to each concept, different categories are distinguished.The most relevant are hyponymy, meronymy and hyperonymy:

– Hyponymy: The specific term used to designate a member of a class; X is a hyponymof Y if X is a (kind of) Y [1]. In the case of verbs, it is known as troponymy; i.e., averb that indicates more precisely the manner of doing something by replacing a verbof a more generalized meaning. For instance, draw defined as write a legal documentor paper is a troponym of write under its gloss “produce a literary work”.

– Meronymy: The name of a constituent part of, the substance of, or a member of some-thing. X is a meronym of Y if X is a part of Y [1]. It is typical in nouns. For instance,the part-whole relation holds between Synsets like chair and {back, backrest}, {seat}and {leg}. Parts are inherited from their superordinates: if a chair has legs, then anarmchair has legs as well.


– Hyperonymy: The generic term used to designate a whole class of specific instances.Y is a hyperonym of X if X is a (kind of) Y [1]. For instance, create verbally is directhyperonym of write of its first gloss. It is the reverse of hyponymy.

Nevertheless, it is worth noting that synonyms are grouped into unordered sets; i.e., thesynonyms are not ordered attending to their degree of synonym or similarity respect to themain concept. Furthermore, nor are they classified according to their linguistic register andhence, vulgar and cultured terms are not distinguished. Finally, the synonymy is not onlybetween terms but also between terms and phrasal verbs; for instance, close up and keep

mum. This case is dealt with in a very irregular way in WordNet [26].Galnet [26] is a version of WordNet adapted to the Galician language. It is integrated into

the EuroWordNet project, that is a version of WordNet improved with an interlingual index(ILI) to European languages. It is a tool that allow us to blind the synsets of the WordNet of aparticular language with the synsets of other languages. It is based on the use of ontologies andhierarchies of domains, avoiding, by this way, false friends that are common in the translationof languages. This neither includes a organization of synsets based on the degree of similaritybut it assumes as objective a best analysis of phrasal verbs. It is under development but thepreliminaries results, for body parts and substances, are available2.

Once the thesaurus has been chosen, the next step is to establish the way to measure thesynonymy between two terms because, as we said before, this value is not provided by thedictionary. We assume the thesis that the degree of synonymy between two given terms canbe calculated using similarity measures [78]. So, let a and b be two terms for calculatingits degree of synonym and A = Synset(a) and B = Synset(b). In [20], we can find severaldefinitions of similarity measures that can be used for our aim:

– Jaccard coefficient: The percentage of synonyms shared by a and b and their union.

SYNJC(a,b) =|A∩B||A∪B|

(3.3)

– Dice coefficient: The proportion between the number of synonyms shared by a and b

and the arithmetic mean of |A| and |B|.

SYNDC(a,b) =|A∩B||A|+|B|

2

=2 · |A∩B||A|+ |B|

(3.4)

2adimen.si.ehu.es/cgi-bin/wei/public/wei.consult.perl


– Cosine coefficient: This is the same as Dice coefficient and with the arithmetic meansubstituted by a geometric mean.

SYNCC(a,b) =|A∩B|√|A| · |B|

(3.5)

– Mutual similarity coefficient: This is also the same as Dice coefficient and with theharmonic mean of number of elements of A and B.

SYNMSC(a,b) =|A∩B|

21|A|+

1|B|

=

|A∩B||A| + |A∩B|

|B|2

(3.6)

– Overlap coefficient: In this case, the denominator is the min of the cardinality of |A|and |B|.

SYNOC(a,b) =|A∩B|

min(|A|, |B|)(3.7)

All these measures are symmetric (the degree of synonymy between a and b is the samethat between b and a). They can be ordered attending to their results3:

SYNJC ≤ SYNDC ≤ SYNCC ≤ SYNMSC ≤ SYNOC (3.8)

We have pointed out that synonymy is, in general, a symmetric relationship although itmay not be symmetric in some particular cases; i.e., if one of the synonymous terms play therole of prototype and the other not. For instance, the synonymy between window and dormer

is not symmetric if we consider window as the prototype (the term that involves any kind ofwindow) and dormer as a particular type because all dormers are windows but not all windows

are dormers.A non-symmetric model of distance measure is the so-called ratio model [84]. This mea-

sure calculates the quotient between the cardinality of the set of synonymous that are sharedby A and B divided by the addition of the cardinalities of the three sets of synonyms: i) the setof synonyms shared by a and b (A∩B); ii) the set of synonyms of a that are not synonymousof b (A−B) with a weight α ≥ 0; iii) the set of synonyms of b that are not synonymous of a

(B−A) with a weight β ≥ 0. The result is the following measure:

SYNRM(a,b) =|A∩B|

|A∩B|+α|A−B|+β |B−A|(3.9)

3The proof for these inequalities can be seen in [20, p. 182]


Values for α and β Obtained Measureα = β = 1 SYNJC

α = β = 1/2 SYNDC

Table 3.8: Equivalences between Ratio Model and Dice and Jaccard coefficients.

This measure generalizes some of the previous ones modifying the values for α and β . Ta-ble 3.8 show them.

Although ratio model is a non-symmetric measure, it is symmetric whether |A| = |B| orwhether α = β . For this reason, this measure seems to be the best one for describing the realbehaviour of synonyms in natural language. However, it entails certain difficulties becausethe cardinalities of A and B depend on the thesaurus dictionary and, therefore, the symmetryor non-symmetry depends on the weights α and β . The main question is how these weightscan be estimated.

A solution for providing a higher degree of synonymy to SYNRM(a,b) than SYNRM(b,a)

if it is the case that a is synonymous of b in the thesaurus but b is not synonymous of a. Hence,the weight of β must be double the weight of α; i.e., β = 2α . The obtaining measure is thefollowing one [84]:

SYNNSRM(a,b) =|A∩B|

|A∩B|+ |A−B|+2 · |B−A|(3.10)

3.3 Basic Inference Pattern

In chapter 2 we presented a proposal of syllogistic reasoning involving fuzzy quantifiers anda way to deal with them. As presuppositions, we assumed fully true statements and crispchaining among the terms. Taking this model as a starting point, our next step is to proposean extension to deal with syllogisms where the chaining is not crisp but fuzzy, understandingthis as those cases where the middle term is played by two similar terms, that can be linkedby a fuzzy inclusion, by a fuzzy overlapping or by synonymy. We only focus on asymmetricsyllogisms, i.e., those which chaining is through a middle term. The chaining of symmetricsyllogisms is based on logic operators, and, therefore, they are out of the aim of this proposal.

The procedure for calculating the value for the quantifier of the conclusion QC is indepen-dent of the strength of the chaining in the middle terms. As we explained, quantifiers onlyrefer to quantity relationships between the terms independently of their semantics. Therefore,any modification in the chaining of the middle term cannot be transferred to the quantifier as

3.3. Basic Inference Pattern 143

NT DT

DT' MTESYN

Figure 3.5: Graphic schema of ACS of Figure I.

it is a semantic relationship independent of the cardinality of the terms. Thus, we propose thatthe conclusion of this type of syllogism must be qualified or weighted by a degree of confi-dence γ . This degree should depend on the degree of synonymy between the involved terms,given that the higher the degree of synonymy between them, the higher the degree of confi-dence on the conclusion. Therefore, we should have γ ∈ [0,1], with γ = 1 when the terms arecompletely equivalent or identical (classical syllogism), and γ = 0 when there is no synonymyrelationship between them (fallacious syllogism). Taking this into account, we propose γ tobe defined as the degree of synonymy between the terms DT and DT’. As a consequence,different quantitative values can be obtained for different synonymy measures (expressions(3.3)-(3.7)). The most adequate one should be chosen for each particular use.

Firstly, we shall consider a syllogism in its basic form; i.e., the one comprising three terms,two premises and a conclusion. The four positions that the middle term can have correspondwith the four Aristotelian figures.

Figure 3.5 shows the graphic representation of Figure I with approximate terms. In acontext E, a syllogistic argument involves an NT and an MT (crisp or fuzzy sets; circle withsolid lines) and two terms (DT and DT’), which are fuzzy sets (circles with dotted lines)related by a SYN relationship that denotes similarity between them.

Figure 3.6 shows the graphic representation of Figure II. In this case, DT and DT’ are thepredicates in both premises.

Figure III is represented in Figure 3.7. In this case, DT and DT’ are the subjects in bothpremises.


NT DT

DT' MTE

SYN

Figure 3.6: Graphic schema of ACS of Figure II.

NT DT

DT' MTE

SYN

Figure 3.7: Graphic schema of ACS of Figure III.

Finally, Figure 3.8 shows the representation of Figure IV. This is the converse of Figure I;i.e., the middle term is the predicate in the first premise and the subject in the second one, butpreserving the same form in the conclusion.

Table 3.9 summarizes the linguistic schemas of the four Aristotelian Figures with syn-onyms in the middle term, where Qi stands for the respective quantifiers of the argument, andDT and DT ′ stands for the similar terms in the role of middle term. It is worth noting that themodification of the position of the middle terms in the premises does not entail any modifica-tion in our underlying thesis (the higher the degree of similarity between terms the higher thedegree of confidence in the conclusion), as this modification is only relevant for calculatingthe quantifier of the conclusion.


NT DT

DT' MTE

SYN

Figure 3.8: Graphic schema of ACS of Figure IV.

Figure I Figure II Figure III Figure IVQ1 DTs are MTs Q1 MTs are DTs Q1 DTs are MTs Q1 MTs are DTsQ2 NTs are DT’s Q2 NTs are DT’s Q2 DT’s are NTs Q2 DT’s are NTs

———————– ———————– ———————– ———————-Q NTs are MTs γ Q NTs are MTs γ Q NTs are MTs γ Q NTs are MTs γ

Table 3.9: The fuzzy chaining version of the four Aristotelian Figures.

3.3.1 A Brief Analysis of the Measures of Synonymy based on Exam-ples

In section 3.2.1, we pointed out that the selection of the measure of synonym depends on thecontext in which the syllogism is used. In this section, we shall introduce some examplesof different situations to discern some rules or indications for the selection of one measureamong others.

In [12, p. 275], the argument of Table 3.10 is proposed as an example of argument wherethe synonymy between the DTs terms, wealthy and rich, is perfect. According to WordNet,their Synsets are the following:

– SYNSET(wealthy)={affluent, flush, loaded, moneyed, wealthy, substantial, rich}

– SYNSET(rich)={rich, affluent, flush, loaded, moneyed, wealthy, substantial}

In this case, for any measure of distance, the result is 1 because SYNSET(wealthy)=SYNSET(rich) and, hence γ = 1. In fact, any of the measures pointed out is valid whensynonymy is total. However, this is not the most frequent case in natural language, but it may


All lawyers are rich peopleNo wealthy people are vagrantsNo lawyers are vagrants

Table 3.10: Syllogism with degree 1 of synonymy in DT.

All denizens are nativeSome citizens are residentSome citizens are native

Table 3.11: Syllogism with a high degree of synonym in DT.

be interesting for scientific fields, where the specific vocabulary is usually univocal ratherpolysemic.

Table 3.11 shows a new example of syllogism of Figure I where the role of DT is takenby the terms denizen and resident. We consider an informal conversation between two friendstalking about democracy and politics.

Firstly, we analyse the number of glosses of each term and if they share some:

– Denizen: i) person who inhabits a particular place; ii) a plant or animal naturalized in aregion.

– Resident: i) someone who lives at a particular place for a prolonged period or who wasborn there; ii) a physician (especially an intern) who lives in a hospital and cares forhospitalized patients under the supervision of the medical staff of the hospital.

In this case, we can see that they share a gloss i) partially, what is more, gloss i) of denizen

is a subclass of gloss i) of resident. The other two glosses of both concepts are completelydifferent (one is about plants and animals while the other is about medicine) and it is notexcessive to assume that this facilitates selection of gloss i) to our context. In this case, wecan draw up a shortlist of synonymy measures between those with the higher values; forinstance, SYNOC and SYNMSC.

So, to apply the measures, we have to choose the synonyms that appears in gloss i). Giventhe relationship between the glosses of both terms explained above, in denizen we focus onsynonyms that fall under the label “direct hyponym” and in resident under the label of “directhyperonym”. It is worth noting that, at this level, resident only has 1 gloss while denizen has30. This can lead us to select the low level measure as there is a big difference between them;


however, if we analyse these glosses in detail, 23/30 are related with concrete geographicalareas such as Europe, Australia, etc. and, hence, they are too specific to be considered inthis context. Therefore, the two previously selected measures SYNOC and SYNMSC will beapplied. Thus, the selected Synsets are the following:

– SYNSET(denizen)={inhabitant, habitant, dweller, denizen, indweller, cottager, cottagedweller, liver, marcher, resident, occupant, occupier, tellurian, earthling, earthman,worldling}

– SYNSET(resident)={resident, occupant, occupier, inhabitant, habitant, dweller, denizen,indweller}

Applying the SYNOC, we obtain the highest degree of synonymy between both terms,which is shown in 3.11:

SYNOC(DE,RE) =|Synset(DE)∩Synset(RE)|

min(|Synset(DE),Synset(RE)|=

=88= 1

(3.11)

i.e., the degree of confidence of the conclusion is γ = 1. For the Mutual similarity coefficient,the other selected, we obtain;

SYNMSC(DE,RE) =|Synset(DE)∩Synset(RE)|

|Synset(DE) + |Synset(DE)∩Synset(RE)||Synset(RE)

2=

=162 + 8

82

= 0.75

(3.12)

that is, γ = 0.75.As we can see, there is a significant difference in the degree of confidence of the conclu-

sion attending to the measure of synonymy that is used. SYNOC emphasizes shared terms withrespect to no shared ones since it is sensitive to the smallest Synset in comparison. As can beobserved, if the smallest Synset is included in the largest one (although there are many othersdifferent synonyms) the result is 1 because fits it with min(Synset(a),Synset(b)). This is aninteresting result as it adequately represents fuzzy inclusion. SynMSC is more restrictive as itweights the Synsets of both terms; although the result obtained is sufficiently high.

Hence, this example shows us that the selection of the Synsets to apply the measure mustbe performed also attending to the number of glosses that the terms share. The criteria for


All dishes are beautifulSome porringers are artworksSome beautiful things are artworks

Table 3.12: Syllogism with a low degree of synonym in DT.

selecting them is necessarily the context in which the syllogism is uttered, but the variabilityin the pattern definition of glosses [26] seems to make this step in one difficulty to automatize.How to solve this question is out of the aims of this research, but it is an open question to afurther development of this version of syllogism.

Table 3.12 shows another example of Figure III, where the middle term is the subject inboth premises. The role of DT is played by the terms dishes and porringers. The context isthe organization of an old house.

The term dish has six glosses;

– a piece of dishware normally used as a container for holding or serving food

– a particular item of prepared food

– the quantity that a dish will hold

– a very attractive or seductive looking woman

– directional antenna consisting of a parabolic reflector for microwave or radio frequencyradiation

– an activity that you like or at which you are superior

and porringer only one;

– a shallow metal bowl (usually with a handle)

As we can see, there is a meaningful difference respect to the number of glosses in bothterms; dish has many more than porringer. Furthermore, dish involves three glosses relatedwith objects in the kitchen and food although in the case of porringer this reference is omitted.For that reason, we propose selecting the measures with the lower values as the similaritybetween both terms seems to be accidental rather structural. Thus, we shall use SYNJC andSYNDC.


Next, we shall show the Synsets of both concepts. Porringer only has one direct hyper-onym while dish has ten direct hyponyms:

– SYNSET(porringer)={porringer, bowl}

– SYNSET(dish)={bowl, butter dish, casserole, coquile, gravy boat, gravy holder, sauce-boat, boat, Petri dish, ramekin, ramequin, serving dish, sugar bowl, watch glass}

Again, we calculate the the value for SYNJC (3.13) and SYNDC (3.14):

SYNJC(PO,DI) =|Synset(PO)∩Synset(DI)||Synset(PO)∪Synset(DI)|

=

=115

= 0.066(3.13)

SYNDC(VO,CI) =2 · |Synset(PO)∩Synset(SI)||Synset(PO)|+Synset(DI)|

=

=216

= 0.125(3.14)

Thus, for SYNJC, γ = 0.66 and for SYNDC, γ = 0.125. These low results are consistentwith our intuition about the syllogism, as the two terms are not so close. Although the conclu-sion of the syllogism is true, it is difficult to accept it as an inference from the premises and,hence, the syllogism has a low degree of reliability.

With these examples we try to show how the context is fundamental for selecting theSynsets that should be considered to calculate the similarity between the terms. It determineswhich glosses are relevant and, from these, we can make a shortlist of similarity measuresto calculate the corresponding γ for the conclusion. However, from the point of view of thisimplementation, the different patterns used to define the glosses make it difficult to deal withthem.

3.3.2 Syllogisms with Weighted Quantified Statements

The fact that the conclusion of fuzzy chained syllogisms is a quantified statement with theform Q A are B [γ], where γ stands for the degree of fulfilment or confidence of the statement,is a major change in the habitual form of the statements involved in a syllogism, since theyare usually assumed to be (fully) true and the usual concern is what types of information canbe inferred from them [88]. To reason with this new form of quantified statements, which we


PR1: Q1 L1,1 are L1,2[γ1]PR2: Q2 L2,1 are L2,2[γ2]. . .

PRN: QN LN,1 are LN,2[γn]C: QC LC,1 are LC,2[γC]

Table 3.13: Broad reasoning schema for weighted quantified statements.

call Weighted Quantified Statements (WQS), it is necessary to incorporate a new procedure tomanage the degree of confidence; hence it is independent of the calculation of the quantifier.

Table 3.13 shows our proposal of syllogistic schema compatible with TGQ developed inchapter 2. The value for Q is calculated according to the usual procedure, independently ofthe value of γ , as the degree of confidence of the conclusion does not modify the value of thequantifier. On the other hand, γC is calculated as min(γ1,γ2) as the premises that constitutethe syllogism are connected by the logical operation AND. This manner of dealing with thedegrees of confidence in the premises is the same as in Mamdani’s and other fuzzy reasoningmodels and uses involving rules with degrees of confidence or certainty factors [14].

The aggregation of the premises is based on and and, therefore, γC must be calculatedaccording to (3.15).

γC =N∧

n=1

γn (3.15)

3.4 Summary

In this chapter we introduce a new form of approximate syllogism involving vagueness notonly in the quantifiers but also in the terms. This inference schema had already been consid-ered in classical crisp syllogism but as a kind of invalid syllogism: the so-called fallacy of fourterms. This appears when the role of middle term is played by two different terms generating,in this way, a non-existent chaining between the extreme terms.

Nevertheless, the terms also can be fuzzy and, although the chaining is not perfect, it canbe strong enough to generate an acceptable argument in the context of everyday reasoning. Inthis case, we assume that the conclusion must be qualified through a degree of confidence γ .It depends directly on the degree of similarity of the middle terms: the higher the degree ofsimilarity, the higher the degree of confidence.

3.4. Summary 151

We explore in depth the similarity in terms of synonymy, a typical semantic relationshipbetween terms. We assume that the higher the number of synonyms that two terms share, thehigher the degree of similarity between them. Using the on line tools, such as WordNet or itsequivalent in Galician language Galnet, we develop a procedure for measuring the synonymyof two terms:

1. Select the gloss(es) of the terms according to the context of the syllogism.

2. Make a shortlist of measures of similarity to be applied depending on the number ofglosses that they share: the higher the number of univocal glosses, the selection thehigher measures of similarity.

3. Select the adequate synset to each concept.

4. Calculate the degree of synonymy of the terms according to the selected measure andmove this value directly to γ .

It is worth noting that this proposal is compatible with our model of syllogistic approachcompatible with TGQ (see chapter 2), since the calculation of the quantifier of the conclusionis independent of its degree of confidence. Furthermore, and following similar models in theliterature if the premises involved in the syllogism are also qualified through a value γ , thedegree of confidence of the conclusion is the result of applying the min operator over the γi ofthe premises because the premises of the syllogism are linked by and operators.

Thus, incorporating vague terms to syllogism is a new extension thereof which helps todeal with more fragments of natural language, as human beings manage a huge number ofvague terms, not only in the quantifiers but also in the other parts of the premises. Never-theless, the context-dependence and the irregular definition of glosses makes it difficult toaddress the automation of this procedure, and the treatment thereof will be proposed as futurework.

On the other hand, obtaining quantified statements qualified by a degree of fulfilment leadsus to consider the Generalized LSO proposed by P. Murinová and V. Nóvak [36] to develop afuzzy LSO and, therefore, obtain a new set of fuzzy immediate inferences. This possibility isproposed as future work.

CHAPTER 4

USE CASES: PROBABILISTIC REASONING

USING SYLLOGISMS

In section 1.3 we explored the equivalence between probabilities and proportional quantifiedstatements; particularly, the use of binary proportional quantified statements as a linguisticway of expressing conditional probabilities; in section 1.2.1, this relationship was analysedfrom the point of view of psychology of reasoning [43] and the syllogistic inference patternwas substituted by probabilistic reasoning. Nevertheless, as pointed out therein, this proposalentails a number of drawbacks, such as its incapability for managing quantifiers other thanproportional ones. In this chapter, we shall adopt the opposite perspective; i.e., to use thesyllogism as a tool for managing conditional probabilities and probabilistic reasoning usingnatural language.

This same idea has already been partially explored in the literature. In [3, 16], in theframework of Interval Syllogistics, a combination between Pattern I (see section 1.1.4) anda Generalized version of Bayes’ Theorem (GBT), characterized by comprising only condi-tional probabilities, is proposed. The result is a hybrid inference schema between syllogismand Bayesian reasoning. However, this is limited to deductive inferences through the chain-ing of the terms involved in the argument. In [27], another proposal for dealing with fuzzyprobabilities is developed using linguistic terms. In this case, fuzzy logic is used to gener-alize the space of classical probability theory ([0,1]) with linguistic labels. This frameworkwas applied to Bayesian Networks (BNs), a probabilistic tool for managing uncertainty thatallows also to make inferences, and a linguistic version of them was developed. Nevertheless,

154 Chapter 4. Use Cases: Probabilistic Reasoning Using Syllogisms

the use of linguistic terms was only devoted to label the probability values, not to the buildingof the whole BN (e.g., the arcs).

In this chapter, we take this a step further proposing the use of natural language (in partic-ular, fuzzy quantified statements) not only for expressing the probabilities of a BN but also forbuilding the entire the reasoning procedure. Syllogisms, given the link pointed out betweenconditional probabilities and proportional quantifiers statements, and using the model that wehave developed, offer us a framework that involves all the necessary tools for this objective:i) precise and vague quantifiers can be managed; ii) the chaining between several terms canbe expressed; iii) our proposal is not reduced to deductive syllogisms but also has a heuristiccharacter; and, iv) there is no restriction in the number of premises and terms that can be used.On the other hand, we shall also check the compatibility between our proposal and the GBT toverify whether we can deal with it, given that we use Interval Syllogistics as the fundamentalbase of our model.

If we adopt the perspective of users, our syllogistic model also shows some interestingaspects to be considered with respect to the probabilistic approaches habitually used to dealwith situations or problems involving uncertainty or vagueness. We can substitute precisenumerical percentages with approximate linguistic quantifiers changing only the quantifier ofthe corresponding premise. In addition, the entire syllogism is built using the same type ofpropositions (binary quantified statements, which everybody is familiar with) and to makeit easier the modelling for non-specialized users; the proper use of probability, in contrast,usually requires certain level of specialized knowledge.

Our main objective is to illustrate how a computational model of syllogism can be usedin other fields than the traditional one (elaborate simple arguments involving only three termsand a few quantifiers) which require the use of natural language argumentation and proba-bilities. We shall illustrate this idea using examples taken from medical diagnoses and legalarguments. We shall use precise cases as, adopting a common idea in fuzzy logic, they mustbe a particular case of fuzzy models; if they are wrong, the fuzzy generalization is not valid.In addition, as explained in chapter 2, the step in our model from precise values to fuzzy onesis simple.

This chapter is organized as follows: section 4.1 describes the main characteristics of thecombination Pattern I with GBT developed in [3, 16] and some inferences using our modelof syllogism; section 4.2 introduces BNs and how their different basic types of topologicalpatterns can be modelled using syllogisms; an example of medical diagnosis is explained;

4.1. Generalized Bayes’ Theorem 155

A1 A2 Ak-1 A3AkP(A2|A1)

P(A1|A2)

P(Ai|Ai+1)

P(Ai+1|Ai)

P(Ak|Ak-1)

P(Ak-1|Ak)

P(A1|Ak)

P(Ak|A1)

Figure 4.1: The chaining of terms described by GBT.

in section 4.3 we illustrate the use of syllogisms in legal argument to avoid probabilisticmistakes; both examples are developed to illustrate through real cases how syllogisms wouldbe built; section 4.4 describes how to build a set of linguistic labels like Interval Syllogisticsproposes (see 1.1.4.2); and, finally, section 4.5 summarizes our main conclusions about thesequestions and the strengths and drawbacks of our proposal.

4.1 Generalized Bayes’ Theorem

An initial proposal for the combination of syllogistic and probability reasoning can be foundin [3, 16]. It is developed in Interval Syllogistics (see 1.1.4) and consists of the combination ofPattern I with the Generalized Bayes’ Theorem (GBT), a version of Bayes’ theorem using onlyconditional probabilities. It addresses arguments involving more than three terms, althoughlimited to an unidirectional chaining among the terms or nodes of the argument. Figure 4.1illustrates the basic structure of GBT, where A1,A2, . . . ,Ak are the nodes of the probabilisticnetwork or the terms of the syllogism. For instance, for k = 4, for calculating P(A4|A1), weneed to know P(A1|A4), P(A1|A2), P(A2|A1), P(A2|A3), P(A3|A2), P(A3|A4) and P(A4|A3).

The mathematical expression of this theorem is shown in (4.1)1.

∀A1, . . . ,Ak,P(A1|Ak) = P(Ak|A1) ∏i=1,k−1

P(Ai|A1)

P(Ai+1|Ai)(4.1)

GBT only uses positive conditional probabilities as the authors assume that the a prioriprobability values that classic Bayes’ theorem needs are unknown. For that reason this versionis qualified as Generalized.

1The proof can be checked in [3, pp. 28-29].


student sport single young childrenstudent [1.000,1.000] [0.700,0.900] [0.000,1.000] [0.850,0.950] [0.000,1.000]sport [0.400,0.600] [1.000,1.000] [0.800,0.850] [0.900,1.000] [0.000,1.000]single [0.000,1.000] [0.700,0.900] [1.000,1.000] [0.600,0.800] [0.050,0.100]young [0.250,0.350] [0.800,0.900] [0.900,1.000] [1.000,1.000] [0.000,0.050]children [0.000,1.000] [0.000,1.000] [0.000,0.050] [0.000,0.050] [1.000,1.000]

Table 4.1: Results of the students’ study about the people in the campus.

sport single

student young

children

[0.8,0.85]

[0.7,0.9][0.6,0.8][0

.9,1]

[0.85,0.95]

[0.25,0.35]

[0.4,0.6][0.7,0.9]

[0.8,0.9]

[0.9,1]

[0.05,0.1][0,0.05]

[0,0.05]

[0,0.05]

Figure 4.2: Example of network for reasoning with Pattern I and GBT.

To illustrate how this proposal works, we shall analyse the example used in [16]. Letus consider the people in a university campus. A group of students in sociology make asmall study analysing the different links among those who are students, single, young, prac-tice sports and have children. To ensure the most reliable values, different students ask thesame questions and obtain slightly different results. To preserve these differences, they arerepresented by means of intervals. Table 4.1 shows the results obtained.

Once the table is completed, they discover that some information is missing. This isrepresented through the interval [0.000,1.000], which means indetermination. Some examplesof this are the conditional probabilities between students and singles or between have children

and practice sport. Figure 4.2 shows the graphical representation of the values in Table 4.1.To obtain the missing values, the framework of Pattern I with GBT can be applied. As

previously stated, GBT does not substitute Pattern I; rather both reasoning schemas are com-plementary. Pattern I can only manage arguments with three term-sets while GBT needs asinitial values the information of arcs that can only be generated using Pattern I. The procedurefor combining both inferences schemas is explained using pseudo-code as follows [3, p. 30]:

4.1. Generalized Bayes’ Theorem 157

while Probability intervals can be further improved dorepeat

the probability intervals cannot be further improved;until Apply Pattern I to generate the missing arcs;repeat

improve the arcs generated by Pattern I;until Apply GBT;

endAlgorithm 1: Pseudo-code for Pattern I and GBT.


Table 4.2: Saturated network of Figure 4.2.

Once the model is applied, the saturated network described in Table 4.2 is obtained. Aswe can see, not only are the values for the missing nodes, such as students and single, im-proved, but also certain values of the initial data, such as students and sport (from [0.7,0.9] to[0.9,0.9]).

Next, we shall apply our model of syllogistic reasoning to illustrate, with some examples,how this problem can also be addressed using fuzzy quantified statements and syllogisms.The first step is to build the premises of the argument, which must include all the availableinformation; i.e., Table 4.1. The form of the premises is the standard one (Q S are P)2, whereS are the nodes of the lines and P the nodes of the columns. This is shown in Table 4.3.

The conclusion has the same form of the premises. For each arch to be calculated, theconclusion has to be modified accordingly. Table 4.4 shows the results obtained, which areidentical to the ones in Table 4.2; keeping very small differences due to the approximatednumerical procedure.

Most of these values (19/25) can be calculated in one step using the argument of Ta-ble 4.3 and changing the corresponding terms in the conclusion. Nevertheless, some of them(6/25) cannot be calculated using this procedure, since there are some incongruities between

2To facilitate the expression of the argument and avoid expressions such us “Q students are players of sports”,the verb to be is substituted in some cases by non-copulative verbs although preserving the meaning of the sentence.


PR1 Between 70% and 90% students play sportsPR2 Between 85% and 95% students are youngPR3 Between 40% and 60% sport players are studentsPR4 Between 80% and 85% sport players are singlePR5 Over 90% of the sport players are youngPR6 Between 70% and 90% single people are sport playersPR7 Between 60% and 80% single people are youngPR8 Between 5% and 10% single people have childrenPR9 Between 25% and 35% young people are studentPR10 Between 80% and 90% young people are sport playersPR11 Over 90% of the young people are singlePR12 Less than 5% of the young people have childrenPR13 Less than 5% of the people with children are youngPR14 Less than 5% of the people with children are singleC Q S are P

Table 4.3: Syllogistic argument for Figure 4.2.


Table 4.4: Results for Figure 4.2 applying syllogisms.

the premises, owing to the fact that our method is global whilst Pattern I with GBT is lo-cal (only arguments with three nodes are considered). We shall illustrate this problem withan example. The arc that links sport players with have children is not compatible with thepremises “[0,0.05] people with children are single” and “[0,0.05] people with children areyoung”. Once this premise has been eliminated, the obtained result, Q = [0,0.1539], is againthe exact result.

In summary, we have illustrated how our proposal of syllogistic system can deal with ar-guments involving only conditional probabilities without restrictions in the number of terms,premises and in the position in the chaining terms. This entails a simplification of the algo-rithm designed for using the GBT, as the division into two steps is not necessary. Furthermore,we obtain more precise results in one single execution in most of the cases, although in some

4.2. Bayesian Networks 159

A B

D E

C

Figure 4.3: Example of BN for causality.

of them more executions are needed to obtain the precise results. In any case, the number ofsteps is less than or equal to those used by the algorithm for GBT.

4.2 Bayesian Networks

BNs are a knowledge representation tool for dealing with domains with uncertainty, codifyingthe information in a graphical structure using probabilities and allowing also to make differenttypes of inferences [30, p. 29]. They are composed of nodes, that represent a set of randomvariables from the domain, and arcs, which link pairs of nodes, showing the direct dependen-cies between them. Figure 4.3 shows an example of a BN, where we can see that nodes A

and B are independent events, as they are not connected between themselves while C, D andE are dependent, as there are arcs that link C with D and E. The only constraint about thesegraphs is that they are directed acyclic graphs (DAGS) (you cannot return to a node simplyby following directed arcs).

Nodes are characterized by being mutually exclusively and exhaustively defined; i.e., eachvariable can only take one single value at a time. The most typical types of nodes are: Boolean(for instance, {True, False}), ordered (for instance, {small, medium big}) and integral (forinstance, {1,2, . . . ,120}). The structure or topology of the network captures the links betweenthe variables. Table 4.5 shows the values that the nodes can take.


Node name Type ValuesA Binary {a,b}B Boolean {True,False}C Boolean {True,False}D Ternary {c,d,e}E Integral {1,2,3,4,5}

Table 4.5: Nodes and values for the BN of Figure 4.3.

C P(D=c|C) P(D=d|C) P(D=e|C)T 0.3 0.5 0.2F 0.7 0.1 0.2

Table 4.6: Nodes and values for the node D of Figure 4.3.

One of the most common uses of BNs is for modelling causality. In these cases, thetopology of the network denotes causality relationships, where some nodes are causes, othersare effects and the arcs indicate the direction from causes to effects. Considering again thenetwork of Figure 4.3, we can say, for instance, that A is a cause of C or E is an effect of C.

This information used to build a BN is usually expressed trough the so-called ConditionalProbability Tables (CPT). Table 4.6 shows an example of CPT for the nodes C and D, wherethe numerical values are always conditional probabilities. It is worth noting the sum of thevalues of each row must be 1 to satisfy the constraint of exhaustiveness.

Regarding reasoning with BNs, they basically allow us to make two types of inferences: i)inferences without evidence, ii) inferences with evidence. Case i) only deals with inferencesfollowing the direction of the arcs (i.e.; from causes to effects in causal BNs) and consistsof in the simple propagation of the values introduced in the BN (a priori probabilities andCPTs). For instance, in the BN of Figure 4.3, let us P(A = a) = 0.9 and P(B = True) = 0.3,a case i) inference is calculating the probability of C; i.e., P(C = True) = 0.0116.

Case ii) involves the so-called evidence. These are the new data that modify the valueof any node of the network. In this case, BNs offer many more possibilities of inferences.This process, known as conditioning (also called probability propagation, inference or beliefupdating), flows through the arcs, although it is not limited to their directions. Basically, thereasoning process consists of calculating the new probability distribution for a set of querynodes given the new observations (or evidence) incorporated into the BN. For instance, in the


A B C

Figure 4.4: Causal Chain network.

BN of Figure 4.3, a case ii) inference is P(D = c|C = T ) = 0.3, which the evidence is (C = T ).Three types of evidences are usually considered:

– Specific evidence: A node takes a particular value; i.e., B = True.

– Negative evidence: The evidence says that the node cannot take one of possible valuesalthough it can take any of the remaining ones; i.e., D = notc.

– Virtual or likelihood evidence: The evidence is incorporated to the BN but it is not fullyreliable.

4.2.1 Topological Types of Bayesian Networks

Next, we shall describe the specific characteristics of BNs when they are used for representingcausality (the parent node is a cause and the child node is an effect). These are the so-calledtopological patterns, which define how the variables are connected and how the evidences inone of them modify the values of the others. There are three basic patterns: causal chain,common causes, and common effects [30, p. 40-41].

4.2.1.1 Causal Bayesian Networks: Causal Chain

Consider a chain between three nodes A,B and C, where A causes B and this C, as shown inFigure 4.4. This type of chain gives rise to conditional independence as P(C|B) = P(C|A,B).In other words, the probability of C depends on the probability of A if and only if there is noevidences of B; in this is the case, the probability of A does not affect C. This idea can besummarized in the following rule:

Rule 1 The evidence is transmitted from an extreme to the another in a causal chain, savethat some intermediate node is instantiated.

The network can be extended to any n number of intermediate nodes between the extremeones.


Step 1 Identifying the number of nodes of the network. They are the terms of thesyllogism.

Step 2 Expressing the a priori probability of the parent nodes as “Q U are X”,where Q denotes the quantifier that expresses the a priori probability (whichcan either be precise, imprecise or fuzzy); U the universe of the discourse orreference; and, X the corresponding node.

Step 3 Expressing the conditional probabilities tables (CPT) as “Q Y are Z”, whereQ denotes the quantifier that express the conditional probability (which canalso be precise, imprecise or fuzzy); Y denotes the occurrence of the givennode or parent node; and, Z the node under consideration or child node; inTGQ, Y denotes the scope and Z the restriction of the quantifier. These state-ments, together with the previous ones, are the premises of the syllogism.

Step 4 Expressing the probability that is aimed to be calculated as the conclusion ofthe syllogism. It follows the same schema; i.e., “Q X are Z”, where Q denotesthe quantifier calculated by our model; X denotes the occurrence of a givennode or the universe of the discourse; and, Z the node under consideration.

Table 4.7: Procedure for building a syllogism for Causal Chain pattern.

To build the syllogism corresponding to this network, the procedure in Table 4.7 must befollowed. We shall illustrate this procedure with the network in Figure 4.4: step 1, there arethree nodes (A, B and C) and, hence, three terms; step 2, (P(A) = 0.3) ≡ “30% elements of

Us are As”; step 3, P(B|A) = 0.05) ≡ “5% As are Bs”; step 4, P(C|A) = 0.20) ≡ “20% As

are Cs”.

4.2.1.2 Causal Bayesian Networks: Common Causes

Consider a network where a node B has two effects, A and C. It is represented by a network ofinverted v-structure; as shown in Figure 4.5. A and C are linked through a common cause B;therefore, evidence of any of the children nodes modifies the remaining ones; i.e., an evidenceabout A modifies the probability value of C through B. If B is instantiated, this is the only onethat determines the value of the other children. This idea can be summarized in the followingrule:


A C

B

Figure 4.5: Common Causes network.

Rule 2 The evidence is transmitted between the children in a common cause network, exceptif that the parent node is instantiated.

The network can be extended to any n number of children nodes.The manner of building the syllogism is the same of causal chain pattern (see Table 4.7)

and we shall illustrate it with the network in Figure 4.5: step 1, the syllogism involves threeterms A, B and C; step 2, the a priori probabilities are of a single node, (P(C) = 0.3)≡ “30%

elements of Us are Cs”; step 3, all the conditional probabilities between B and A and B andC must be also included; for instance, “10% Cs are As”, “60% Cs are Bs”, etc.; step 4, theconclusion, e.g. “5% Us are As” or “30% Cs are As”.

4.2.1.3 Causal Bayesian Networks: Common Effects

Consider a network where a node B (effect) has two causes, A and C. It is represented bya network v-structure; as shown in Figure 4.6. The parent nodes, A and C, are independent,which means that evidence of any of them does not modify the value of the other. Neverthe-less, they become conditionally dependent if the common effect is instantiated. The followingrule summarizes this idea:

Rule 3 The evidence is not transmitted between the parents in a common effect network,except if the child node is instantiated.

The network can be extended to any n number of parent nodes.The strategy for establishing the syllogism is the same of the previous models, although an

additional consideration is needed. The procedure is shown in Table 4.8 and we shall illustrateit with the network in Figure 4.6: step 1, the three terms involved, A, B and C; step 2, the a

priori probabilities of the parent nodes, “20% Us are As” and “30% elements of Us are Cs”,and their indepdence, P(A,C) = P(A) ·P(C); i.e., (P(A) ·P(C) = 0.2 ·0.3 = 0.06)≡ “6% Us


A C

B

Figure 4.6: Common Effects network.

Step 1 Identifying the terms of the syllogism from the nodes of the network.

Step 2 The a priori probabilities of the parent nodes must be included. In addition,the independence between these nodes must also be explicitly included as theirjoint probability according to the independence rule. This value is expressedthrough a conjunction in the predicate of the premise.

Step 3 The conditional probabilities between the parent and child nodes of theCPTs.

Step 4 The conclusion preserves the standard form; i.e., “Q X are Z”.

Table 4.8: Procedure for building a syllogism for Common Effects pattern.

are As and Cs”; step 3, “60% As are Bs”, “70% Cs are Bs”, etc.; step 4, “10% Us are Cs”,etc.

Once we have defined the structure and the different forms that the networks can adopt,the next step is to describe the reasoning procedure. BNs allows different types of inferences,which are analysed in the following section.

4.2.2 Types of Inferences with Bayesian Networks

The use of BNs for reasoning enables us to execute different types of inferences. Dependingon their direction, three basic types are defined [30, p. 34]:

– Predictive: It goes from causes to effects; i.e., given evidence about a cause, the newprobabilities of each effect is calculated. It follows the direction to the arcs.


– Diagnostic: It goes from effects to causes; i.e., given evidence about an effect, theprobability of each cause is calculated. It follows the opposite direction to the arcs; it isthe opposite of the previous one.

– Intercausal (explaining away): It is the reasoning about mutual causes of a commoneffect. Thus, given two causes initially independent, if we have new evidence aboutone of the causes and the effect, the probability of the other cause is modified. Inother words, two causes initially disconnected become dependent by the common effectupdated by new evidence.

Diagnostic and predictive inferences can be executed in all networks described in sec-tion 4.2.1 while Intercausal inference can only be applied in common causes pattern, since itis devoted to analysing the dependency generated between two previously independent nodes.There is an additional pattern of inference that can be mentioned, although we do not con-sider it as a basic one as it is a combination between two of the basic ones: diagnostic andpredictive. It is the so-called combined inference.

Figure 4.7, taken from [30, p. 34], illustrates these four types of reasoning using a BN withfive binary nodes with an X-structure (a combination between inverted v and v structures),which results from the combination of common causes and common effects patterns. Thedotted ellipses denote the nodes with the new evidence and the grey ellipses the queries thatmust be solved.

To illustrate how each one of these types of inference works and, how they can be modelledusing syllogisms, we shall use an example of medical diagnosis of lung cancer also took fromKorb’s book [30, p. 30].

4.2.3 BNs in Medical Diagnosis: Lung Cancer

Let us suppose the fictitious hospital Princeton-Plainsboro, where the following situation hap-pens:

A patient has been suffering from shortness of breath (called dyspnea) and visitsthe doctor, worried that he has lung cancer. The doctor knows that other diseases,such as tuberculosis and bronchitis, are possible causes, as well as lung cancer.She also knows that other relevant information includes whether or not the patientis a smoker (increasing the chances of cancer and bronchitis) and what sort of


A B

D E

C

Intercausal

Query

Evidence

A B

D E

C

Dir

ecti

on o

f re

ason

ing

Diagnostic

Query

Query

Evidence

A B

D E

C

Direction of reasoning

Predictive

Query

Query

Evidence

A B

D E

C

Combined

Query

Evidence

Query

Evidence Evidence

Figure 4.7: Types of Bayesian reasoning.

Node name Type ValuesPollution Binary {Low,High}Smoker Boolean {True,False}Cancer Boolean {True,False}Dyspnoea Boolean {True,False}X-ray Binary {Positive,Negative}

Table 4.9: Types and values of the nodes of BN for the lung cancer problem.

air pollution he has been exposed to. A positive X-ray would indicate eithertuberculosis or lung cancer.

With the information included in this wording it is possible to identify the variables andvalues described in Table 4.9 and the BN described in Figure 4.8, where the correspondingCTP are also stated.

In the following subsections, we shall analyse the different types of inference and howthey must be expressed as syllogisms.


Pollution Smoker

Cancer

XRay Dyspnoea

S

T

F

P(C=T|P,S)

0.05

0.02

P

H

H

TL

FL

0.03

0.001

P(P=L)

0.90

P(P=H)

0.10

P(S=T)

0.30

P(S=F)

0.70

C

T

F

P(X=P|C)

0.90

0.20

P(X=N|C)

0.10

0.80

C

T

F

P(D=T|C)

0.65

0.30

P(D=F|C)

0.35

0.70

P(C=F|P,S)

0.95

0.98

0.97

0.999

Figure 4.8: A BN for the lung cancer problem.

4.2.3.1 Predictive Reasoning

The oncologist, J. Wilson, wants to know the a priori probability that the patient has lungcancer, X-Ray positive and dyspnoea. From the point of view of the inference to be executedin the BN, this is a simple propagation without additional evidence using any of the explainedpatterns (see section 4.2.1).

The elaboration of a syllogism comprises several steps. As previously stated, the premisesmust include the a priori probabilities of the parent nodes over a particular universe and theCPTs. In this case, given that the network has an X-structure with five binary nodes, there aretwo premises for the a priori probabilities of the root node, one for including the independencebetween them (which is calculated as the product of the a priori probability of each node), andeight for including all the conditional probabilities of the CPTs. In the example of Figure 4.8,


PR1 90% of people live with LPPR2 30% of people are SPR3 3% of peopole live under HP and are SPR4 5% of people that live under HP and are S have CPR5 2% of people that live under HP and are not S have CPR6 3% of people that live under LP and are S have CPR7 0.1% of people that live under LP and are not S have CPR8 90% of people with C have positive XPR9 20% of people without C have positive XPR10 65% of people with C have DPR11 30% of people without C have DC Q S are P

Table 4.10: Syllogistic argument about lung cancer.

the nodes Pollution and Smoker are the root nodes; independent, which entails3:

P(PH&ST |E) = P(PH) ·P(ST ) = 0.1 ·0.3 = 0.03 (4.2)

P(PH&SF |E) = P(PH) ·P(SF) = 0.1 ·0.7 = 0.07 (4.3)

P(PL&ST |E) = P(PL) ·P(ST ) = 0.9 ·0.3 = 0.27 (4.4)

P(PL&SF |E) = P(PL) ·P(SF) = 0.9 ·0.7 = 0.63 (4.5)

Table 4.10 shows the syllogistic argument corresponding the Figure 4.8. PR1 and PR2denotes the a priori probability of the independent events live with low pollution air and be

smoker and PR3 the joint probability of both; and PR4 to PR11 is the information of theCPTs. It is worth noting that in this example, since all variables are binary or Boolean, it isonly necessary to include half of the information of the CPTs (with one of the values) becausethe other can be directly derived from these by the exhaustiveness of the network.

The conclusion comprises the information that is aimed to obtain; i.e. the query of the BN(see Figure 4.7). In general, the subject term is the evidence and the predicate term is the querynode. The calculation of having cancer is the direct application of common causes pattern,while the calculation of positive X-Ray and Dyspnoea entails the combination of commoncauses pattern with causal chain one.

3LP stands for “low pollution”, HP stands for “high pollution”, S stands for “smoker”, C stands for “cancer”, Xstands for “X-Ray” and D stands for “dyspnoea”.


Conclusion Syllogism BNQ people have cancer 0.011630 1.16%Q people have positive X-Ray 0.208141 20.8%Q people have Dyspnoea 0.304074 30.4%

Table 4.11: Conclusions for simple propagation.

Conclusion Additional Premise Syllogism BNQ S have C 0.032000 3.20%Q S have D 30.7% D are S 0.031116 31.1%Q people live with HP air have D 89.2% of people with D live with LP 0.310152 31.0%Q people C have X 0.900000 90.0%

Table 4.12: Conclusions for predictive reasoning with evidence.

Given that in this case our queries are about a priori probabilities, the subject term of theconclusion is the universe of reference. Table 4.11 summarizes the conclusions obtained withour model compared with the obtained from the BN4.

Another possibility in the predictive reasoning is the inference with new evidences aboutthe nodes; for instance, the probability of have a positive X-Ray given that the patient issmoker. This is expressed in syllogistic terms as “Q smokers have positive X-Ray”, accordingto the said above.

In this case, an additional consideration must be taken into account. If the new evidence isabout a parent node of the query node taken; no additional premises are required; for instance,for the conclusion “Q smokers have cancer” it suffices to consider the basic set of premises.In other case, the inverse statement of the conclusion must be added as an additional premise;for instance, in order to conclude “Q smokers have dyspnoea” the statement “Q patients withdyspnoea are smokers” must be added as an additional premise. If this information is notincorporated into the argument, the quantifier of the conclusion will be more imprecise, al-though consistent in any case with the precise result. It is worth noting that through an iterativeprocedure (such us the one explained in section 4.1) it is possible to specify, iteration by it-eration, the obtained result. Table 4.12 shows some results with the additional premises thatmust be added with each conclusion, if it is the case.

4The BN was modelled with NETICA, a graphical tool for modelling BNs.


Conclusion Additional Premise Syllogism BNQ D have C 0.024861 2.49%Q D live with HP air 31.0% D are PH 0.101950 10.2%Q people with X positive have D 21.7% D have positive X 0.304000 30.4%Q people with X negative are S 77.8% S have negative X 0.295075 29.5%

Table 4.13: Conclusions for diagnostic reasoning.

4.2.3.2 Diagnostic Reasoning

Traditionally, syllogism was devoted exclusively to deductions which means, in terms of BNs,predictive inferences. However, as we have pointed out in chapter 2, our model has also aheuristic character and, hence, other type of inferences can be managed. In this section, weshall illustrate its behaviour respect to diagnostic reasoning.

Diagnostic inferences always require evidence; therefore, the inverse one from the con-clusion must be added as an additional premise in almost all cases. Only when the querynode (the predicate-term of the conclusion) is the parent of the evidence (the subject-term) isthis additional premise not necessary. As in the predictive reasoning with evidences, if thesepremises are not included, the final results obtained are more imprecise.

Table 4.13 illustrates some examples. As can be observed, the results of the syllogismcoincide with the BN. Diagnostic inference can be applied in any of the inference patternsexplained in section 4.2.1.

4.2.3.3 Intercausal Reasoning

Intercausal reasoning or explaining away is mainly applied in inferences that use a commoncauses pattern. For instance, in our example, given the evidence of being smoker and havingcancer, calculate the probability of being exposed to low pollution air.

This type of inference requires using at least two items of evidence instead of a singleone. This condition can be easily expressed in the syllogism. Thus, given that two itemsof evidence are simultaneously required, the linguistic way for expressing it is through anand: i.e., a conjunction in logic terms. Since our model can manage this type of Booleanoperations, both items of evidence must appear in the subject of the conclusions linked by anand. In the example that we are analysing, it is only possible between the nodes pollution,smoker and cancer. Given that the nodes under consideration are immediate, no additionalpremises are demanded. Table 4.14 shows four conclusions using intercausal pattern.

4.3. Probabilistic Reasoning in Legal Argumentation 171

Conclusion Additional Premise Syllogism BNQ PH with C are S 0.517241 51.7%Q PH without C are S 0.293512 29.4%Q S without C are PL 0.901860 90.2%Q S with C are PH 0.156250 15.6%

Table 4.14: Conclusions for combined reasoning.

4.2.3.4 Combined Reasoning

Combined reasoning is defined as the combination of predictive and diagnostic reasoning. Aswe have explained above, we do not consider this model as a basic one.

From the point of view of syllogism, this type of reasoning can present certain problemswith the inclusion of the inverse of the conclusion as an additional premise as two directionsof the inference are executed at the same time. With the current definition for syllogisms thatwe manage, inconsistencies between the premise and the conclusion appear in some cases.

Given that the other three basic reasoning schemas can be effectively expressed in termsof syllogisms, the development of this type of inference will be addressed as future work.

4.3 Probabilistic Reasoning in Legal Argumentation

Not all probabilistic reasoning is straightforward to be modelled and executed through BNs.We shall focus on legal argumentation, as it is a discipline in which the arguments in naturallanguage are fundamental and can determine the guilt or innocence of a person.

We shall illustrate the usefulness of our model of syllogistic reasoning in this field analys-ing the so-called Prosecutor’s fallacy [83]. This consists of a bad use of probabilities instatistical reasoning and it is typically used by the prosecution, to argue the guilt of the ac-cused. Several cases of this type of argument has been documented [68], such us the SallyClark case5 or the well-known case of People vs. Collins [30, pp.18-19], which we shallconsider in what follows:

In 1964 an interracial couple was convicted of robbery in Los Angeles, largelyon the grounds that they matched a highly improbable profile, a profile which fitwitness reports. In particular, the two robbers were reported to be a particular set

5A document from the Royal Statistical Society about this case can be consulted inhttp://www.rss.org.uk/uploadedfiles/documentlibrary/744.pdf.


Characteristics ProbabilitiesA man with a moustache 1/4Who was black and had a beard 1/10And a woman with a ponytail 1/10Who was blonde 1/3The couple was interracial 1/1000And were driving a yellow car 1/10

Table 4.15: Characteristics and probabilities of the suspicious couple.

of characteristics and the prosecution suggested that these had the probabilitiesshown in Table 4.15.

The prosecutor calls an instructor of mathematics who, assuming that all theseevents are mutually independent, applies the “product rule” obtaining that thecouple were innocent was no more than 1/12000000. The jury was persuadedand the couple convicted.

Now, we shall express this case using a syllogism and applying our model to estimate theprobability of the couple being guilty. Our universe the population of Los Angeles and theinformation of Table 4.15 the used one to build the premises of the argument. The conclusionis the frequency of a couple like the one described by the witness in the set of couples in LosAngeles; i.e., “Q people are a black man with a moustache and beard and a blonde woman witha ponytail and an interracial couple driving and yellow car”. Table 4.16 shows the argumentwhich result is Q = [0,1]; i.e., inconclusive. It shows that the set of premises used to buildthis syllogism is not enough to achieve a relevant conclusion.

It is obvious that these premises are not independent but that there are some clear links be-tween them. For instance, if a man has a moustache, the probability of having beard changes,as very few people with beard do not have moustache. In [30, p. 19-20], a new analysis ofthis problem attending to the conditional probabilities that links these premises among themis explained. For instance, PR2 entails PR1 and PR2; PR3 and PR4 entails PR5, which giveus a probability of innocence around 1/3000; which means that any couple that satisfies theseproperties is guilty. The next step is to calculate the a priori probability of a random coupleis being guilty of robbery. Attending to the population of Los Angeles and being generous,there are 1,625,000 possible couples and, hence, the probability of being a guilty couple is1/1625000. With this new information, we obtain the argument of Table 4.17.

4.3. Probabilistic Reasoning in Legal Argumentation 173

PR1 25% people have moustachePR2 10% people are black and had beardPR3 10% people are women with a ponytailPR4 33% people are blonde womenPR5 0.1% people are interracial couplesPR6 10% people are driving a yellow carPR7 100% man with a mustache and black with beard and

a woman with a ponytail and blonde and an interracial coupleand drive and yellow car are guilty

C Q man with a moustache and black with beard anda woman with a ponytail and blonde and an interracial coupleand drive and yellow car are not guilty

Table 4.16: Syllogism for the case People vs Collins.

PR1 0.03% interracial couple are a man with a mustacheand black with beard and a woman with a ponytail and blonde

PR2 0.000006% couple of Los Angeles are guiltyPR3 100% interracial couple are coupleC Q man with a moustache and black with beard and

a woman with a ponytail and blonde are guilty

Table 4.17: Syllogism with the conditional probabilities of People vs Collins.

The application of our model produces again Q = [0,1]; i.e., the result is still inconclusive.Although applying Bayesian reasoning a result of 0.002 chance of culpability is obtained [30,p. 20], which does not mean that 99.8% is the best probability of innocence either. Thesyllogistic argument reduces this probability to a total indetermination as it only considers theends; i.e., with the available information, the set of couples who committed the crime. For aresult where Q = [0,0] means that no couple committed the crime; for Q = [1,1] means thatall couples committed the crime, given that the set under considerations has a single coupleas member, the culpability is total.

Thus, we can conclude that the conclusions of a syllogistic argument are more careful,less committed than probabilistic inferences as they only consider the ends. This may be avirtue or a drawback, depending on the context and its purpose. For instance, in this type ofuse, it is more favourable for the accused.

The basic inferences shown in this chapter were executed using SEREA (the softwarelibrary developed by ourselves referred to in chapter 2). Currently, it us under development


and only BNs with less than five nodes can be managed by it. More complicated networks (forinstance, the BN developed in the example of section 4.2.3) require a handmade formalizationto be performed before its computational execution. In this case, we have developed a smallprogramme in Octave for the optimization step.

4.4 Qualitative Version using Linguistic labels

We illustrated the behaviour of our syllogistic model for managing some examples of prob-abilistic reasoning in two fields where many of the probabilities can be expressed using lin-guistic quantifiers instead of exact probabilities. For instance, instead of saying “90% of thepopulation is exposed to low air pollution” we say “the vast majority of population is exposedto low air pollution” or “0.000006% couple in Los Angeles are guilty” is substituted by “al-most no couple in Los Angeles is guilty”. Thus, using our method, a linguistic version ofBayesian reasoning can be built, where the probabilities are defined through a set of linguis-tic labels and the results are also expressed and approximated to acceptable values for thoselinguistic quantifiers. The obtained results will not be totally precise but they will be moreinterpretable and therefore more use-oriented than precise ones from the point of view of theirusability.

An initial extension to introduce linguistic labels is the proposal analysed in section 1.1.4,where a way of introducing linguistic quantifiers into fuzzy syllogism without using a versionof the LSO is explained. As has also been mentioned, the main differences between bothapproaches is that the LSO is based on the truth properties of the quantified statements, whilethe use of lattices of linguistic labels is based on the properties of order and certainty of thequantifiers; there are no considerations about the truth value of the statements. Our proposalis fully compatible with the way linguistic quantifiers are introduced into Interval Syllogistics.Therefore, the procedure described in section 1.1.4.2 can be directly applied to our proposal.

It is true that all the examples described involve crisp values instead of fuzzy ones. Thisfact had two motivations: i) the examples are taken from a book about classic probabilisticreasoning [30]; ii) in this way, we illustrate how the crisp case can be perfectly managed withour model assuming the principle that the fuzzy case must include a crisp one as a particularcase.

4.5. Summary 175

4.5 Summary

We have proposed three fields, where the information is usually expressed using natural lan-guage, for applying our framework for syllogistic reasoning: i) inferences using only condi-tional probabilities; ii) inferences using BNs; iii) arguments using other type of probabilisticreasoning. All the explained examples show a correct behaviour of our model, although theelaboration of the syllogistic argument needs certain considerations for its correct definition.There are summarized in Table 4.18.

We emphasize that the expression of probabilistic problems in syllogisms contributes tomaking the problem more interpretable. The premises and the conclusion have the same form(Q A are B) and new evidence can be directly added in the subject of the conclusion. Onthe other hand, it can deal simultaneously with precise and fuzzy quantifiers, which is a verytypical situation in the real word. In addition, it is also relevant to note that the interval orfuzzy set obtained for the quantifier of the conclusion also denotes how complete and precisethe information of the premises is: the smaller the size of the result, the higher the precision ofinformation of the premises. Its compatibility with the use of lattice of linguistic quantifiersallows us to easily generate a kind of linguistic Bayesian reasoning; which, for the fieldsconsidered (medical diagnosis and legal argument) is simpler, since the building of a BN andits associated conditional probabilities usually demands the collaboration of an expert.

We have also illustrated how to express a problem by a syllogism and how it can help usto check the validity of our arguments. We show that a bad argument can have highly negativeconsequences, such as an wrongful conviction in the case of People vs. Collins shown insection 4.3.

By way of conclusion, it is also relevant to note that we have proposed new applicationfields for syllogisms different to its traditional use, showing in this way new possibilities forit. In addition, new types of inferences, different from the deductive ones, were included as apart of syllogistic inferences, something new in this type of argument.


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Conclusions and Future Work

The starting hypothesis of this research was the development of a model for dealing withsyllogistic arguments involving fuzzy quantifiers, more than three terms or more than twopremises. To achieve this aim, several objectives have been defined which we analyse below.

The first objective was an in-depth review of different proposals of syllogistic models inthe literature. Two main interpretations for the quantified statements that constitute syllogismswere identified: i) as assertions that define a particular quantity relationship between two sets;ii) as sentences whose depth form is a conditional proposition where the subject and thepredicate respectively correspond with the antecedent and the consequent of the conditionaland the quantifier denotes how strong is the link between them.

We called interpretation i) set-based interpretation, and it assumes an ontology based onsets. Thus, quantified statements are assertions whose purpose is to describe a fact denotingthe link between two sets of elements, the subject and the predicate terms, and, the quanti-fier denotes the cardinality of that relationship. Interpretation ii), which we call conditional

interpretation, assumes an ontology based on individuals and properties. Thus, quantifiedstatements are complex propositions with a conditional form, where the subject is the an-tecedent and the predicate is the consequent. In this way, the fact is described in terms of itstruth conditions and not as a description, and the role of the quantifier is to denote how strongis the link between the antecedent and the consequent, which can denote probability, belief ormodality.

The analysis of both proposals led us to some relevant conclusions for the subsequent stepsof this work. Conditional interpretation allows us to transform syllogisms into argumentsof first order logic and to use well known reasoning patterns such as Modus Ponens (likeQualified Syllogistics proposes). On the other hand, it also has a direct link with probabilisticreasoning insomuch as quantifiers are interpreted in terms of probabilities. Nevertheless,

178 CONCLUSIONS AND FUTURE WORK

from the point of view of the aim of this research, it shows two relevant drawbacks: i) alogical analysis of the statement to identify the conditional form under the natural languagesurface is required; and, ii), the most important one, it does not have a clear way for managingquantifiers different to the proportional ones (“most”, “few”, etc.) such us absolute (“five”,“around ten”, etc.) or comparative (“around double”, “less than half”, etc.), which are alsovery common in natural language. For both questions, set-based interpretation shows a betterbehaviour as it allows us to preserve the natural language form and manage the different typesof quantifiers defined by TGQ. Despite this, the equivalence between proportional quantifiedstatements and conditional probabilities is fruitful, as we shall explain, for using syllogismsto deal with some problems involving probabilities.

On the other hand, we also adopt Aristotelian Syllogistics as the classical referentialframework, as it is the base of this type of reasoning and its compatibility with TGQ alsohas been proved. It is worth noting that in all but two of the analysed syllogistic systemsbased on set-based interpretation their compatibility with Aristotelian Syllogistics is explic-itly analysed. The exceptions are Interval Syllogistics, which has been analysed in this thesisand is compatible, and Fuzzy Syllogistics, which is not compatible with Aristotelian Syllogis-tics and, in addition presents some other meaningful problems for its use. All these objectiveswere achieved in chapter 1.

The next step was the development of a syllogistic model based on the set-based interpre-tation incorporating the definitions of TGQ and making flexible the inference pattern. Firstly,we assumed Venn Diagrams (with more or fewer modifications) as the adequate knowledgerepresentation tool for syllogisms, as all syllogistic models analysed except the fuzzy ones.They offer us a clear representation of the structure of the relationships involved in a syllo-gistic argument in terms of disjoint subsets and allow us to assign to each one the cardinality(fuzzy or not) denoted by the quantifier. At this step, we introduced algebraic methods torewrite every premise of the syllogism as an inequation whose values are the bounds of thequantifier. If the quantifier is precise, the bounds match up; if it is imprecise, the bounds de-note an interval; if it is fuzzy, the bounds are a four-tuple (the typical notation for a trapezoidalfunction) used together with α-cuts. In this way, we are interpreting syllogistic inferences as amathematical optimization problem, which opened up the possibilities for using them not onlyas deductions but as a heuristic procedure for finding the conclusion that is most compatiblewith the restrictions or constraints defined as premises. Thus, we have divided the process intothree phases: i) dividing the universe of discourse into disjoint sets; ii) defining sentences as

CONCLUSIONS AND FUTURE WORK 179

systems of inequations; iii) selecting the optimization method that should be applied in eachcase in order to resolve the reasoning process. When the quantifier that must be optimized isan absolute one, the Simplex method can be used directly, while if it is a proportional one,linear fractional programming techniques must be adopted. These results satisfy the secondand third objectives initially defined in this research.

The interpretation of this reasoning process in terms of a mathematical optimization prob-lem led us to a syllogism that not only makes deductions but is also valid for heuristic pro-cedures and non-deductive inferences, where the conclusion denotes the proposition morecompatible with the premises, being or not a logical consequence of them stricto sensu. Fur-thermore, our proposal also overcomes most of the habitual limitations identified in the othersyllogistic systems:

– It can deal with arguments with no limitations in the number of premises and terms.

– It admits the combination of different types of quantifiers in the same argument.

– It is not limited to a set of inference patterns where the position of the middle term(s) ispredefined but that it can appear in any position.

As we have also stated, vagueness can also appear in terms. For this reason, once the sys-tem for dealing with vague quantifiers was developed, we addressed the problem of vaguenessin terms. We mainly focus on the middle term, since any minimum change on it may generatea fallacy. Thus, we started exploring the literature on similarity measures between terms andwe identified three approaches: i) similarity as fuzzy inclusion, ii) similarity as fuzzy overlap-ping and iii) similarity as synonymy; each one of them with its proper context of use. Afteranalysing them, we concluded that the first two are developed sufficiently but the third onehas some points to be improved. With these results, we satisfy the fourth objective of thiswork, the study of different alternatives to measure similarity between terms.

To improve the use of synonymy as a similarity measure, we adopted the following idea:the higher the number of synonyms two terms share, the higher the degree of similarity be-tween them. In addition, we used a pragmatic definition of synonymy: two words are syn-onyms if they appear as synonyms in a thesaurus. As thesauruses, we chose WordNet forEnglish and Galnet for Galician language, as they are very complete and provide additionalinformation to synonymy, such us glosses, which are relevant for our goals. Thus, using thedata supplied by them, we developed an approach divided into two steps: i) calculating the


proportion of glosses shared by the terms and, ii) calculating the number of synonyms sharedby these glosses. Step i) provides a criterion for choosing the similarity measure from amongthe standard ones (the Jaccard, Dice, Cosine, Mutual similarity, Overlap and Ratio model co-efficients) following the principle that the higher the proportion of glosses shared, the higher

the similarity measure chosen; step ii) to provide the synonyms that must be used in the cal-culation. Using this information, we built an extension of our syllogistic system, using thesame principle as Mamdani’s reasoning model, where a chaining based on similarity is di-rectly translated into a degree of confidence in the conclusion. In this way, we satisfied thefifth objective of this thesis: we built a reasoning schema with approximate chaining wherenot all the arguments are fallacies and the approximated nature of synonymy (that is also verycommon in natural language) is translated into the conclusion.

Syllogism was traditionally devoted to arguments of a very specific type. Nonetheless,the equivalence between quantified statements and conditional probabilities opens up the pos-sibility for studying an analogous link in the field of reasoning. This constitutes, precisely,the sixth objective of this work. Specifically, we explored how diverse examples of Bayesianreasoning can be modelled using syllogisms.

First, we analysed a probabilistic reasoning model that only uses conditional probabilities(the described one as the combination of Pattern I of Interval Syllogistics with GeneralizedBayes Theorem). Our proposal achieved the same results as this model, employing fewer stepsand using the standard form of syllogism without additional distinctions. This contributes tomaking the arguments more easily understandable for non-specialized users and to clearlyascertaining all the premises that a syllogism comprises. In addition, another relevant consid-eration is that a highly unspecific quantifier in the conclusion means that the information ofthe premises is not sufficient to infer a more informative conclusion.

Other case analysed is Bayesian Networks (BNs). We illustrated how a case of medicaldiagnosis can be expressed in syllogistic terms. As we have explained, the premises of thesyllogism express the a priori probabilities and the corresponding conditional probability ta-bles. In the necessary cases, this set of premises can be complemented with the correspondingadditional premises according to the type of inference. The conclusion, in any case, is the in-formation that we attempt to obtain, where the subject-term comprises the evidence of the BNand the predicate the query node.

The final case analysed is an example of Bayesian reasoning in legal argumentation. Wefocus on how an erroneous modelling through Bayesian reasoning of a situation can be dis-


covered using a syllogistic formulation. While the Bayesian argument was incriminating andconclusive, the syllogism was inconclusive as the information of the premises, as is explained,was not sufficient to support the conclusion. Thus, our model of syllogism can be used as atool (for non-specialized users) that contributes to avoiding unjust verdict, as several caseshave been documented.

In addition, we believe that the use of syllogisms also clarifies the reasoning with proba-bilities in other aspects:

– Syllogism allows us to use quantified statements to express probabilities. We have ex-plained how a priori probabilities, conditional probabilities and independence betweenevents must be explained and preserving for all these cases the standard form: Q S are

P.

– Our model not only deals with precise probabilities but also with imprecise ones. Wehave developed the examples using precise probabilities to illustrate that our modelincludes them as a particular case. The extension to imprecise or fuzzy quantifiers iseasy: by substituting the precise values for intervals or fuzzy sets.

– Rendering implicit knowledge explicit. One of the critical points of BNs is the mod-elling of the problem, that is usually complex for many users due to its numerical nature.The use of the syllogism requires all the information used to be make explicit; the moreprecise the premises are, the more precise the quantifier of the conclusion will be.

– Syllogism is not only valid for deductions but also for other types of reasoning. We haveshown how different types of inferences, such as abductive or intercausal ones, can beexecuted using our syllogistic framework. This breaks away the traditional conceptionof syllogism and opens up new possibilities for this type of argument since only a smallpart of human reasoning is purely deductive.

– It is true that the calculation procedure in the syllogism can be opaque, but it is veryclear what the premises and the conclusion are. This fact facilitates the communicationand the understanding of the problem for non-specialized users.

The computational implementation of our model forms part of the seventh objective. Wehave developed a software library, SEREA, which was used in the basic examples (with fewerthan five terms). It is still under development to improve its behaviour. The most complicated


examples (arguments with five terms or more) were implemented in Octave, also obtainingsuitable results.

To conclude, we shall evaluate the satisfaction of the initial hypothesis according to theresults obtained for the defined objectives. All of them have been satisfactorily achieved;therefore, the hypothesis of developing a syllogistic framework for dealing with argumentsinvolving fuzzy terms and quantifiers as well as more flexible inference patterns has beenaccomplished.

Future work

The achieved results in this research have suggested us some interesting topics for the futurework.

Our immediate challenge is to improve our model of syllogistics to deal with other typesof quantifiers other than the binary ones. We have focused mainly on binary quantifiers asthey are, perhaps, the most common ones in ordinary reasoning and an equivalence betweenbinary quantified statements and probabilities can be established. However, there are otherquantifiers defined in the TGQ, such us the ternary (for instance, “there twice as many dogsas cats and parrots”), that also appear in natural language and, therefore, also can be used forreasoning. This idea allows us to define a new type of syllogism comprising premises withdifferent arity: binary, ternary, etc.

Another interesting task is to explore more complex inferences in BNs. In this research,the basic patterns of BNs (the three topological structures with the three basic types of infer-ence) have been explored. Thus, our next step will be to analyse more complex patterns inorder to verify whether the syllogism preserves a consistent behaviour; for instance, networkswith more than five nodes, diagnostic or predictive inferences with more than one item ofevidence, intercausal reasoning with more than two causes, etc.

Approximate chaining syllogism also allows subsequent developments. We have mainlyexplored the concept of synonymy as the support for a similarity measure between the chain-ing terms. Nevertheless, other possibilities can be contemplated, such us using the concepts ofanalogy or antonymy instead of synonymy, which may generate new models of approximatedchaining. On the other hand, the study of the context deserves special attention in this type ofinference, since it can determine, for instance, the gloss used in each argument. In addition, it


is one of the major challenges for the further evolution of Computing with Words6. We havedealt with this problem giving a principal role to the user, who has to assume all the deci-sions of the construction in the arguments, but the challenge in computational intelligence isto manage this automatically.

We also identify computational tasks. The first step is to improve the current version ofthe SEREA library for dealing with all the linguistic quantifiers that we have addressed in thisresearch, such as similarity or comparative ones, and syllogistic inferences involving morethan five terms. A preliminary version has been implemented in Octave, which indicates thatthe migration to SEREA should be straightforward. As a medium-term task, taking SEREAas starting point, we shall aim to develop a software package similar to available ones fordealing with BNs, such us NETICA. Thus, we propose to create a graphical interface to buildthe knowledge representation. Over this graphic model, the corresponding syllogism will begenerated, incorporating the corresponding premises according to Table 4.18, displaying it tothe user. A further possibility is the opposite way; i.e., from a syllogism introduced by theuser, to generate a graphical representation. In addition, the graphical interface also allow usto include the management of linguistic labels described in Interval Syllogistics (1.1.4), wherethe user, for instance, could select between different numbers of predefined linguistic labelsand obtain results approximated to them or even group together some of these labels. Fi-nally, the approximate chaining syllogism also will be broadly developed, using the graphicalinterface to inform the user about different similarity relationships and their results.

6This problem was analysed in the panel session How to deal with Context in Computing with Words? in the lastedition 2013 IEEE International Conference on Fuzzy Systems (FuzzIEEE2013) held in Hyderabad in July 7-10.

Bibliography

[1] Princeton University “About WordNet.” WordNet, 2010.

[2] SEREA, March 2012.

[3] S. Amarger, D. Dubois, and H. Prade. Constraint propagation with impreciseconditional probabilities. In Proceedings of the 7th Conference on Uncertainty in

Artificial Intelligence, pages 26–34, Los Angeles, CA, July 13-15 1991.

[4] Aristotle. Prior and posterior analytics. Clarendom Press, Oxford, 1949.

[5] Aristotle. Aristotelis Politica. Oxford University Press, Oxford, 1957.

[6] Aristotle. Categories and De Interpretatione. Clarendon Press, 1975.

[7] Aristotle. Prior Analytics. Hackett, Indianapolis, 1989.

[8] J. Barwise and R. Cooper. Generalized quantifiers and natural language. Linguistics

and Philosophy, 4:159–219, 1981.

[9] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, TheEdinburgh Building, Cambridge, UK, 2004.

[10] Alberto J. Bugarin and S. Barro. Fuzzy reasoning supported by petri nets. IEEE

Transactions on Fuzzy Systems, 2(2):135–150, 1994.

[11] Alberto J. Bugarin and S. Barro. Reasoning with truth values on compacted fuzzychained rules. IEEE Transactions on Systems, Man, and Cybernetics, Part B:

Cybernetics, 28(1):34–46, 1998.

[12] Irving M. Copi. Introduction to logic. MacMillan, New York, 1994.

186 Bibliography

[13] J. Corcoran. Ancient Logic and Its Modern Interpretations. Synthese HistoricalLibrary. Springer, 1974.

[14] O. Cordón, M. J. del Jesus, and F. Herrera. A proposal on reasoning methods in fuzzyrule-based classification systems. International Journal of Approximate Reasoning,20:21–45, 1999.

[15] A. P. Dempster. A generalization of bayesian inference. Journal of the Royal Statistical

Society. Series B (Methodological), 30(2):pp. 205–247, 1968.

[16] D. Dubois, L. Godo, R. López de Mántaras, and H. Prade. Qualitative Reasoning withImprecise Probabilities. Journal of Intelligent Information Systems, 2:319–363, 1993.

[17] D. Dubois and H. Prade. On fuzzy syllogisms. Computational Intelligence,4(2):171–179, 1988.

[18] D. Dubois, H. Prade, and J-M. Toucas. Intelligent Systems. State of the Art and Future

Directions, chapter Inference with imprecise numerical quantifiers, pages 52–72. EllisHorwood, Great Britain, 1990.

[19] L. Eciolaza, M. Pereira-Fariña, and G. Trivino. Automatic linguistic reporting indriving simulation environments. Applied Soft Computing, 13(9):3956 – 3967, 2013.

[20] S. Fernández-Lanza. Una Contribución al Procesamiento Automático de la Sinonimia

Utilizando Prolog. PhD thesis, University of Santiago de Compostela, 2002.

[21] Tim Fernando and Hans Kamp. Expecting many, 1996.

[22] P. T. Geach. Reason and Argument. University of California Press, 1976.

[23] B. Geurts. Reasoning with quantifiers. Cognition, 86:223–251, 2003.

[24] G. Girard. Synonymes françois et leur différents significations et le choix qu’il faut faire

pur parler avec justesse. Veuve d’Houry, Paris, 1749.

[25] I. Glöckner. Fuzzy Quantifiers: A Computational Theory. Springer, Berlin, 2006.

[26] X. Gómez Guinovart, X. M. Gómez Clemente, A. González Pereira, and V. TaboadaLorenzo. Galnet: Wordnet 3.0 do galego. LinguaMÁTICA, 3(1):61–67, 2011.

Bibliography 187

[27] J. Halliwell and Q. Shen. Linguistic probabilities: theory and application. Soft

Computing, 13:169–183, 2009.

[28] H. Hoijer. Language in Culture: Comparative studies of cultures and civilizations,chapter The Sapir Whorf hypothesis, pages 92 – 105. Comparative studies of culturesand civilizations. University of Chicago Press, 1954.

[29] W. Kneale and M. Kneale. The Development of Logic. Clarendon Press, Oxford, 1968.

[30] K.B. Korb and A.E. Nicholson. Bayesian Artificial Intelligence. Chapman &Hall/CRC, 2004.

[31] I. Landau. Ways of being rude. Technical report, Ben Gurion University, 2006.

[32] Y. Liu and E. E. Kerre. An overview of fuzzy quantifiers (ii). reasoning andapplications. Fuzzy Sets and Systems, 95:135–146, 1998.

[33] J. Lukasiewicz. Aristotle’s Syllogistic. The Clarendon Press, Oxford, 1957.

[34] J. Malczewski. Ordered weighted averaging with fuzzy quantifiers: Gis-basedmulticriteria evaluation for land-use suitability analysis. International Journal of

Applied Earth Observation and Geoinformation, 8:270–277, 2006.

[35] George A. Miller. Wordnet: A lexical database for english. Communications of the

ACM, 38(11):39–41, 1995.

[36] P. Murinová and V. Novák. Analysis of generalized square of opposition withintermediate quantifiers. Fuzzy Sets and Systems, (0):In press, 2013.

[37] P. Murinová and V. Novák. A formal theory of generalized intermediate syllogisms.Fuzzy Sets and Systems, 186:47–80, 2012.

[38] Wallace A. Murphree. Numerically Exceptive Logic: A Reduction of the Classical

Syllogism. Peter Lang, New York, 1991.

[39] Wallace A. Murphree. The numerical syllogism and existential presupposition. Notre

Dame Journal of Formal Logic, 38(1):49 – 64, 1997.

[40] V. Novák. On fuzzy type theory. Fuzzy Sets and Systems, 149:235–273, 2005.

188 Bibliography

[41] V. Novák. A comprehensive theory of trichotomous evaluative lingusitic expressions.Fuzzy Sets and Systems, 159:2939–2969, 2008.

[42] V. Novák. A formal theory of intermediate quantifiers. Fuzzy Sets and Systems,159:1229–1246, 2008.

[43] M. Oaksford and N. Chater. Bayesian Rationality. The probabilistic approach to human

reasoning. Oxford University Press, Oxford, 2007.

[44] Barbara H. Partee. Many quantifiers. In J. Powers and K. de Jong, editors, Proceedings

of the 5th Eastern States Conference on Linguistics, pages 383–402, Columbus, 1989.Ohio State University.

[45] J. Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press,2000.

[46] C. S. Peirce. Collected Papers of Charles Sanders Peirce. Harvard University Press,Cambridge MA, 1931 – 1958.

[47] C. S. Peirce. Collected Papers. Cambridge. Harvard University Press, Cambridge, MA,1933.

[48] M. Pereira-Fariña. Una revisión del concepto de intencionalidad en un marcoaproximado. In Actas del XVI Congreso Español de Tecnologías y Lógica Fuzzy

(ESTYLF2012), pages 435–441, 17-19 de Septiembre, Cuencas Mineras (Mieres -Langreo), Asturias, España, 2008.

[49] M. Pereira-Fariña. A revision about the concept of intentionality in an approximateframework. Mathware and Soft Computing, 16:5–15, 2009.

[50] M. Pereira-Fariña, F. Díaz-Hermida, and A. Bugarín. An analysis of reasoning withquantifiers within the Aristotelian syllogistic framework. In Proc. 2010 International

Fuzzy Conference on Fuzzy Systems (FUZZ-IEEE 2010), pages 635–642, July, 18-23,Barcelona, Spain, 2010.

[51] M. Pereira-Fariña, F. Díaz-Hermida, and A. Bugarín. On the analysis of set-basedfuzzy quantified reasoning using classical syllogistics. Fuzzy Sets and Systems, 214:83– 94, 2013.

Bibliography 189

[52] M. Pereira-Fariña, L. Eciolaza, and G. Trivino. Quality assessment of linguisticdescription of data. In Actas del XVI Congreso Español de Tecnologías y Lógica Fuzzy

(ESTYLF2012), pages 608 – 613, 1 - 3 de Febrero, Valladolid, España, 2012.

[53] M. Pereira-Fariña and A. Sobrino. Recursos lingüístico-formales para el silogismoaproximado. Submitted to XVII Congreso Español de Tecnologías y Lógica Fuzzy

(ESTYLF2014).

[54] M. Pereira-Fariña and A. Sobrino. Una propuesta de silogismo encadenadoaproximado. In C. Martínez Vidal, J. L. Falguera, J. M. Sagüillo, Víctor M. Verdejo,and M. Pereira-Fariña, editors, Actas del VII Congreso de la Sociedad de Lógica,

Metodología y Filosofía de la Ciencia en España, pages 705 –713, 18-20 de Julio,Santiago de Compostela, España, 2012.

[55] M. Pereira-Fariña, A. Sobrino, and A. Bugarín. A proposal of fuzzy chained syllogismbased on the concept of synonymy. In Proc. 2013 International Fuzzy Conference on

Fuzzy Systems (FUZZ-IEEE 2013), pages 1–8, July, 7-10, 2013 - HICC, Hyderabad,India, 2013.

[56] M. Pereira-Fariña, Juan C. Vidal, F. Díaz-Hermida, and A. Bugarín. A fuzzy syllogisticreasoning schema for generalized quantifiers. Fuzzy Sets and Systems, 234:79 – 96,2014.

[57] M. Pereira-Fariña, Juan C. Vidal-Aguiar, P. Montoto, F. Díaz-Hermida, and A. Bugarín.Razonamiento silogístico aproximado con cuantificadores generalizados. In Actas del

XVI Congreso Español de Tecnologías y Lógica Fuzzy (ESTYLF2012), pages 701–706,1 - 3 de Febrero, Valladolid, España, 2012.

[58] S. Peters and D. Westersthål. Quantifiers in language and logic. Clarendom Press,Oxford, 2006.

[59] P. Peterson. Higher quantity syllogisms. Notre Dame Journal of Formal Logic,26(4):348–360, 1985.

[60] P. L. Peterson. On the logic of “few”, “many”, and “most”. Notre Dame Journal of

Formal Logic, 20(1):155–179, 1979.

190 Bibliography

[61] P. L. Peterson. Distribution and proportion. Journal of Philosophical Logic,24(2):193–225, 1995.

[62] P. L. Peterson. How to use aristotelian logic to formalize reasonings expressed inordinary language. Technical Report FS-96-04, AAAI, Menlo Park, California, 1996.

[63] P. L. Peterson. Intermediate Quantities. Logic, linguistics, and Aristotelian semantics.Ashgate, Alsershot. England, 2000.

[64] N. Pfeifer. Argumentation in Theorie und Praxis: Philosophie und Didaktik des

Argumentierens, chapter Contemporary syllogistics: Comparative and quantitativesyllogisms, pages 57–71. LIT, Wien, 2006.

[65] G. Polya. Mathematics and Plausible Reasoning: Induction and Analogy in

Mathematics V. 1. Princenton University Press, 1990.

[66] W.V.O. Quine. Word and Object. MIT Press, Cambridge - Mass, 1960.

[67] W.V.O. Quine. From a Logical Point of View, chapter Two dogmas of empiricism,pages 20–46. Har, Cambridge - Mass, 1980.

[68] D. Kim Rossmo. Criminal Investigative Failures. CRC Press Taylor & Francis Group,2009.

[69] B. Russell. From Frege to Gödel, chapter Letter to Frege, pages 124–125. HarvardUniversity Press, 1967 edition, 1902.

[70] B. Russell. The Problems of Philosophy. Williams and Nogate, London - UnitedKingdom, 1957.

[71] Daniel G. Schwartz. Dynamic reasoning with qualified syllogisms. Artifical

Intelligence, 93:103–167, 1997.

[72] G. Shafer. A Mathematical Theory of Evidence. Princenton University Press,Princteton, NJ, 1976.

[73] C. E. Shannon and W. Weaver. The Mathematical Theory of Communication.University of Illinois Press, Urbana, 1979.

Bibliography 191

[74] A. Sobrino-Cerdeiriña and S. Fernández-Lanza. Synonymy from a computational andphilosophical perspective. In Proceedings eighth international conference IPMU,volume II, Madrid, 2000. CSIC.

[75] A. Sobrino-Cerdeiriña, S. Fernández-Lanza, and J. Graña-Gil. Soft Computing in Web

Information Retrieval. Models and Applications, chapter Access to a Large Dictionaryof Spanish Synonyms: A Tool for Fuzzy Information Retrieval, pages 299–316.Springer-Verlag, 2006.

[76] S. Solt. Understanding Vagueness. Logical, Philosophical and Linguistic Perspectives,chapter Vagueness in quantity: Two case studies from a linguistic perspective, pages157–174. Collegue Publications, 2011.

[77] F. Sommers. The Logic of Natural Language. Clarendon Press, Oxford, 1982.

[78] K. Sparck Jones. Synonymy and Semantic Classification. Edinbürg University Press,1986.

[79] M. Spies. Syllogistic Inference under Uncertainty. Psychologie Verlags Union, 1989.

[80] A. Tarski. The semantic conception of truth: and the foundations of semantics.Philosophy and Phenomenological Research, 4(3):pp. 341–376, 1944.

[81] B. Thompson. Syllogisms using “few”, “many”, and “most”. Notre Dame Journal of

Formal Logic, 23(1):75–84, 1982.

[82] B. Thompson. Syllogisms with statistical quantifiers. Notre Dame Journal of Formal

Logic, 27(1):93–103, 1986.

[83] William C. Thompson and Edward L. Schumann. Interpretation of statistical evidencein criminal trials: The prosecutor’s fallacy and the defense attorney’s fallacy. Law and

Human Behavior, 11(11):167–187, 1987.

[84] A. Tversky. Features of similarity. Psychological Review, 84(4):327–352, 1977.

[85] J. Venn. On the diagrammatic and mechanical representation of propositions andreasonings. Philosophical Magazine and Journal of Science, 9(59):1–18, 1880.

[86] D. Westersthål. Aristotelian syllogisms and generalized quantifiers. Studia Logica,48(4):577–585, 1989.

192 Bibliography

[87] A. N. Whitehead and B. Russell. Principia Mathematica, volume 3. CambridgeUniversity Press, Cambridge, 1910-1913.

[88] Ronald R. Yager. Reasoning with fuzzy quantified statements: Part II. Kybernetes,15:111–120, 1986.

[89] Ronald R. Yager. Quantifier guided aggregation using owa operators. International

Journal of Intelligent Systems, 11(1):49–73, 1996.

[90] L. A. Zadeh. A Computational Approach to Fuzzy Quantifiers in Natural Language.Computer and Mathematics with Applications, 8:149–184, 1983.

[91] L. A. Zadeh. The Role of Fuzzy Logic in the Management of Uncertainty in ExpertSystems. Fuzzy Sets and Systems, 11:199–227, 1983.

[92] L. A. Zadeh. Aspects of Vagueness, chapter A Theory of Commonsense Knowledge,pages 257–296. D. Reidel, Dodrecht, 1984.

[93] L. A. Zadeh. Syllogistic reasoning in fuzzy logic and its applications to usuality andreasoning with dispositions. IEEE Transactions On Systems, Man and Cybernetics,15(6):754–765, 1985.

[94] L. A. Zadeh. Test-score semantics for natural languages and meaning representation viapruf. In George J. Klir and Bo Yuan, editors, Fuzzy sets, fuzzy logic, and fuzzy systems,pages 542–586. World Scientific Publishing Co., Inc., River Edge, NJ, USA, 1996.

List of Figures

Fig. 1.1 Classical LSO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Fig. 1.2 Modern LSO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Fig. 1.3 Primary diagram of the term-sets A and B. . . . . . . . . . . . . . . . . . . 32Fig. 1.4 Venn-Peirce diagram for “All A are B and Some B are not A”. . . . . . . . . 32Fig. 1.5 Ferio syllogism and its representation using Venn-Peirce diagrams. . . . . . 33Fig. 1.6 LSO with “most” and “many”. . . . . . . . . . . . . . . . . . . . . . . . . . 34Fig. 1.7 5-Quantity LSO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Fig. 1.8 General schema for generating the LSO for fractional quantifiers. . . . . . . 38Fig. 1.9 Venn Diagram subsections. . . . . . . . . . . . . . . . . . . . . . . . . . . 40Fig. 1.10 Graphic representation of “between 30% and 50%”. . . . . . . . . . . . . . 53Fig. 1.11 Graphic representation of “30%”. . . . . . . . . . . . . . . . . . . . . . . . 53Fig. 1.12 Graphic representation of “around 40%”. . . . . . . . . . . . . . . . . . . . 54Fig. 1.13 The structure ordered of linguistic quantifiers. . . . . . . . . . . . . . . . . 55Fig. 1.14 Antonymy relationship between “few” and “most” in a interval scale. . . . . 56Fig. 1.15 Graphic representation of Pattern I. . . . . . . . . . . . . . . . . . . . . . . 59Fig. 1.16 Graphic representation of Pattern II. . . . . . . . . . . . . . . . . . . . . . . 60Fig. 1.17 Graphic representation of Pattern II’. . . . . . . . . . . . . . . . . . . . . . 61Fig. 1.18 Graphic representation of Pattern III. . . . . . . . . . . . . . . . . . . . . . 62Fig. 1.19 Graphic representation of Pattern III’. . . . . . . . . . . . . . . . . . . . . . 63Fig. 1.20 Graphic representation of Intersection/product Syllogism. . . . . . . . . . . 69Fig. 1.21 Graphic representation of Multiplicative Chaining syllogism. . . . . . . . . 69Fig. 1.22 Graphic representation of MPR chaining syllogism. . . . . . . . . . . . . . 70Fig. 1.23 Graphic representation of A. Conjunction and A. Disjunction. . . . . . . . . 72Fig. 1.24 Generalized 5-Quantity LSO. . . . . . . . . . . . . . . . . . . . . . . . . . 79

194 List of Figures

Fig. 2.1 Division of the referential universe for the case of S= 3 properties and 8= 23

elements in the partition P′k, k = 1, . . . ,8. P1 = P′5∪P′6∪P′7∪P′8;P2 = P′3∪P′4∪P′7∪P′8;P3 = P′2∪P′4∪P′6∪P′8. . . . . . . . . . . . . . . . . . . . . . . . . . 109

Fig. 2.2 Graphical representation of the solution obtained for QC for the example inTable 2.11, SUPQC = [0.2,1] and KERQC = [0.42,1]. . . . . . . . . . . . . 118

Fig. 2.3 Graphical representation of QC with 11 α−cuts. . . . . . . . . . . . . . . . 119Fig. 2.4 Graphical representation of the solution obtained for QC for the example in

Table 2.11 with the additional premise PR4. . . . . . . . . . . . . . . . . . 120Fig. 2.5 Qc of Example 2.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Fig. 2.6 Graphical representation of the solution obtained for QC for the example in

Table 2.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Fig. 3.1 Counter example of ACS when DT ′ is included in DT . . . . . . . . . . . . . 131Fig. 3.2 Counter example of ACS when DT is included in DT ′. . . . . . . . . . . . . 132Fig. 3.3 α indicates the overlapping between A and A′. . . . . . . . . . . . . . . . . 133Fig. 3.4 The term write on WordNet. . . . . . . . . . . . . . . . . . . . . . . . . . . 139Fig. 3.5 Graphic schema of ACS of Figure I. . . . . . . . . . . . . . . . . . . . . . . 143Fig. 3.6 Graphic schema of ACS of Figure II. . . . . . . . . . . . . . . . . . . . . . 144Fig. 3.7 Graphic schema of ACS of Figure III. . . . . . . . . . . . . . . . . . . . . . 144Fig. 3.8 Graphic schema of ACS of Figure IV. . . . . . . . . . . . . . . . . . . . . . 145

Fig. 4.1 The chaining of terms described by GBT. . . . . . . . . . . . . . . . . . . . 155Fig. 4.2 Example of network for reasoning with Pattern I and GBT. . . . . . . . . . . 156Fig. 4.3 Example of BN for causality. . . . . . . . . . . . . . . . . . . . . . . . . . 159Fig. 4.4 Causal Chain network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Fig. 4.5 Common Causes network. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Fig. 4.6 Common Effects network. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Fig. 4.7 Types of Bayesian reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . 166Fig. 4.8 A BN for the lung cancer problem. . . . . . . . . . . . . . . . . . . . . . . 167

List of Tables

Tabla 1.1 Four standard categorical statements. . . . . . . . . . . . . . . . . . . . . 24Tabla 1.2 Distribution in categorical statements. . . . . . . . . . . . . . . . . . . . . 24Tabla 1.3 Basic immediate inferences. . . . . . . . . . . . . . . . . . . . . . . . . . 26Tabla 1.4 Conversion inferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Tabla 1.5 Obversion inferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Tabla 1.6 Contraposition inferences. . . . . . . . . . . . . . . . . . . . . . . . . . . 27Tabla 1.7 Example of Aristotelian syllogism. . . . . . . . . . . . . . . . . . . . . . . 29Tabla 1.8 Figures of Aristotelian Syllogistics. . . . . . . . . . . . . . . . . . . . . . 30Tabla 1.9 Moods of Aristotelian Syllogistics. . . . . . . . . . . . . . . . . . . . . . . 30Tabla 1.10 Ferison syllogism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Tabla 1.11 Aristotelian counterexample. . . . . . . . . . . . . . . . . . . . . . . . . . 31Tabla 1.12 Equivalence between “few” and “most”. . . . . . . . . . . . . . . . . . . . 35Tabla 1.13 Moods of Intermediate Syllogistics. . . . . . . . . . . . . . . . . . . . . . 39Tabla 1.14 Algebraic interpretation of intermediate categorical statements. . . . . . . 40Tabla 1.15 TTI syllogism and proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Tabla 1.16 Recasting A, E, I and O as exception statements. . . . . . . . . . . . . . . 43Tabla 1.17 General form of exception statements. . . . . . . . . . . . . . . . . . . . . 44Tabla 1.18 Equivalent relationships between exception statements. . . . . . . . . . . . 45Tabla 1.19 Equivalent relationships between exception quantifiers. . . . . . . . . . . . 47Tabla 1.20 Datisi syllogism with exception propositions. . . . . . . . . . . . . . . . . 48Tabla 1.21 Moods of Exception Syllogistics. . . . . . . . . . . . . . . . . . . . . . . 49Tabla 1.22 TTI syllogism combining intermediate and exception quantifiers. . . . . . 50Tabla 1.23 Correspondence between the linguistic terms, their intervals in the scale

and the values when a = 0.2 and b = 0.4. . . . . . . . . . . . . . . . . . . 56

196 List of Tables

Tabla 1.24 Pattern I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Tabla 1.25 Pattern II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Tabla 1.26 Pattern II’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Tabla 1.27 Pattern III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Tabla 1.28 Pattern III’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Tabla 1.29 Position of the middle term in the Aristotelian figures and Pattern I. . . . . 65Tabla 1.30 Behaviour of Pattern I with respect to the six moods in Figure I. . . . . . . 65Tabla 1.31 An example of Zadeh’s fuzzy syllogism. . . . . . . . . . . . . . . . . . . . 67Tabla 1.32 Zadeh’s general fuzzy syllogistic scheme. . . . . . . . . . . . . . . . . . . 67Tabla 1.33 Yager’s immediate inferences. . . . . . . . . . . . . . . . . . . . . . . . . 68Tabla 1.34 Intersecion/Product syllogism. . . . . . . . . . . . . . . . . . . . . . . . . 68Tabla 1.35 Multiplicative Chaining syllogism. . . . . . . . . . . . . . . . . . . . . . . 69Tabla 1.36 Major Premise Reversibility Chaining syllogism. . . . . . . . . . . . . . . 70Tabla 1.37 Antecedent Conjunction/Disjunction schemas. . . . . . . . . . . . . . . . 71Tabla 1.38 Antecedent Conjunction/Disjunction examples. . . . . . . . . . . . . . . . 71Tabla 1.39 The position of the middle term in Aristotelian Figures and Zadeh’s MC

and MPR patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Tabla 1.40 Behavior of Zadeh’s patterns with respect to Figure I. . . . . . . . . . . . . 73Tabla 1.41 Probabilistic expression of “all”, “none”, “some”, “some. . . not”, “few” and

“most”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Tabla 1.42 Inference schema of syllogism under a probabilistic approach. . . . . . . . 86Tabla 1.43 Inference schema of syllogism AII (Figure I) under a probabilistic approach. 87Tabla 1.44 Chaining/Property Inheritance based on Support Logic. . . . . . . . . . . . 92Tabla 1.45 Intersection/Product based on Support Logic. . . . . . . . . . . . . . . . . 92Tabla 1.46 Antecedent Conjunction based on Support Logic. . . . . . . . . . . . . . . 93Tabla 1.47 Consequent Conjunction based on Support Logic. . . . . . . . . . . . . . . 93Tabla 1.48 Interrelations across several level of modifiers. . . . . . . . . . . . . . . . 94Tabla 1.49 Example of qualified syllogism. . . . . . . . . . . . . . . . . . . . . . . . 95Tabla 1.50 Canonical form of qualified syllogism. . . . . . . . . . . . . . . . . . . . . 96

Tabla 2.1 Definition of some binary quantifiers according to TGQ. . . . . . . . . . . 105Tabla 2.2 Inference scheme of broad syllogism. . . . . . . . . . . . . . . . . . . . . 106Tabla 2.3 Aristotelian moods expressed using broad inference pattern. . . . . . . . . 106

List of Tables 197

Tabla 2.4 ETD (Figure II) from Intermediate Syllogistics expressed using broadinference pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Tabla 2.5 AI- (Figure III) from Exception Syllogistics expressed using broadinference pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Tabla 2.6 Pattern II from Interval Syllogistics expressed using broad inference pattern. 107Tabla 2.7 Intersection/product from Fuzzy Syllogistics expressed using broad

inference pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Tabla 2.8 Definitions of interval crisp Generalized Quantifiers Q = [a,b] . . . . . . . 110Tabla 2.9 Syllogism with quantifiers of exception. . . . . . . . . . . . . . . . . . . . 114Tabla 2.10 Syllogism with quantifiers of exception, logic and additional premises. . . 115Tabla 2.11 Syllogism with five terms and five premises. . . . . . . . . . . . . . . . . . 116Tabla 2.12 Zadeh’s Intersection/product syllogism. . . . . . . . . . . . . . . . . . . . 121Tabla 2.13 Zadeh’s Consequent Disjunction syllogism with different conclusion. . . . 122Tabla 2.14 Peterson’s BKO syllogism. . . . . . . . . . . . . . . . . . . . . . . . . . . 124Tabla 2.15 Zadeh’s Intersection/product syllogism. . . . . . . . . . . . . . . . . . . . 124Tabla 2.16 AII mood of Figure III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Tabla 3.1 Syllogism with vagueness in the DT. . . . . . . . . . . . . . . . . . . . . . 130Tabla 3.2 Syllogism with approximate chaining. . . . . . . . . . . . . . . . . . . . . 132Tabla 3.3 Syllogism where the additional premise for Spanish and European. . . . . 132Tabla 3.4 Mamdani’s Approximated Chaining schema. . . . . . . . . . . . . . . . . 133Tabla 3.5 Syllogism with overlapping between DT and DT’. . . . . . . . . . . . . . 134Tabla 3.6 Syllogism with DT and DT’ synonyms. . . . . . . . . . . . . . . . . . . . 134Tabla 3.7 Examples of syllogisms with the terms interchanged. . . . . . . . . . . . . 135Tabla 3.8 Equivalences between Ratio Model and Dice and Jaccard coefficients. . . . 142Tabla 3.9 The fuzzy chaining version of the four Aristotelian Figures. . . . . . . . . 145Tabla 3.10 Syllogism with degree 1 of synonymy in DT. . . . . . . . . . . . . . . . . 146Tabla 3.11 Syllogism with a high degree of synonym in DT. . . . . . . . . . . . . . . 146Tabla 3.12 Syllogism with a low degree of synonym in DT. . . . . . . . . . . . . . . . 148Tabla 3.13 Broad reasoning schema for weighted quantified statements. . . . . . . . . 150

Tabla 4.1 Results of the students’ study about the people in the campus. . . . . . . . 156Tabla 4.2 Saturated network of Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . 157Tabla 4.3 Syllogistic argument for Figure 4.2. . . . . . . . . . . . . . . . . . . . . . 158

198 List of Tables

Tabla 4.4 Results for Figure 4.2 applying syllogisms. . . . . . . . . . . . . . . . . . 158Tabla 4.5 Nodes and values for the BN of Figure 4.3. . . . . . . . . . . . . . . . . . 160Tabla 4.6 Nodes and values for the node D of Figure 4.3. . . . . . . . . . . . . . . . 160Tabla 4.7 Procedure for building a syllogism for Causal Chain pattern. . . . . . . . . 162Tabla 4.8 Procedure for building a syllogism for Common Effects pattern. . . . . . . 164Tabla 4.9 Types and values of the nodes of BN for the lung cancer problem. . . . . . 166Tabla 4.10 Syllogistic argument about lung cancer. . . . . . . . . . . . . . . . . . . . 168Tabla 4.11 Conclusions for simple propagation. . . . . . . . . . . . . . . . . . . . . . 169Tabla 4.12 Conclusions for predictive reasoning with evidence. . . . . . . . . . . . . 169Tabla 4.13 Conclusions for diagnostic reasoning. . . . . . . . . . . . . . . . . . . . . 170Tabla 4.14 Conclusions for combined reasoning. . . . . . . . . . . . . . . . . . . . . 171Tabla 4.15 Characteristics and probabilities of the suspicious couple. . . . . . . . . . 172Tabla 4.16 Syllogism for the case People vs Collins. . . . . . . . . . . . . . . . . . . 173Tabla 4.17 Syllogism with the conditional probabilities of People vs Collins. . . . . . 173Tabla 4.18 Guidelines for building a syllogism. . . . . . . . . . . . . . . . . . . . . . 176

approximate syllogistic reasoning: a … · de la silogística aristotélica cabe destacar que la...

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